Exact bipartite Tur\'an numbers of large even cycles

TL;DR

This paper determines the exact maximum number of edges in bipartite graphs avoiding large even cycles, confirming a conjecture, improving bounds, and establishing a tight edge condition for consecutive even cycles.

Contribution

It proves the exact bipartite Turán number for large even cycles and establishes a tight edge condition for consecutive even cycles, confirming and strengthening previous conjectures.

Findings

Exact formula for bipartite Turán numbers of large even cycles

Tight edge condition for consecutive even cycles in bipartite graphs

Improved bounds over previous results

Abstract

Let the bipartite Tur\'an number $ex(m,n,H)$ of a graph $H$ be the maximum number of edges in an $H$-free bipartite graph with two parts of sizes $m$ and $n$, respectively. In this paper, we prove that $ex(m,n,C_{2t})=(t-1)n+m-t+1$ for any positive integers $m,n,t$ with $n\geq m\geq t\geq \frac{m}{2}+1$. This confirms the rest of a conjecture of Gy\"{o}ri \cite{G97} (in a stronger form), and improves the upper bound of $ex(m,n,C_{2t})$ obtained by Jiang and Ma \cite{JM18} for this range. We also prove a tight edge condition for consecutive even cycles in bipartite graphs, which settles a conjecture in \cite{A09}. As a main tool, for a longest cycle $C$ in a bipartite graph, we obtain an estimate on the upper bound of the number of edges which are incident to at most one vertex in $C$. Our two results generalize or sharpen a classical theorem due to Jackson \cite{J85} in different ways.

Equations

$$e(G)>\left\{\begin{array}[]{ll}(n-1)(t-1)+m,&if~{}m\leq 2t-2;\\ (m+n-2t+3)(t-1),&if~{}m\geq 2t-2.\end{array}\right.$$

$$ex(m,n,C_{2t})\le (t-1)n+m-t+1.$$

$$ex(m,n,C_{2t})=(t-1)n+m-t+1.$$

$$ex(m,n,C_{2t})\leq\left\{\begin{array}[]{ll}(2t-3)\cdot[(mn)^{\frac{t+1}{2t}}+m+n],&\mbox{if }t\mbox{ is odd};\\ (2t-3)\cdot[m^{\frac{t+2}{2t}}n^{\frac{1}{2}}+m+n],&\mbox{if }t\mbox{ is even}.\end{array}\right.$$

$$ex(m,n,C_{2t})\le (t-1)n+c_t\cdot m^2.$$

$$ex(m,n,C_{2t})\le (t-1)n+d_t\cdot m^{1+\frac{1}{[t/2]}}.$$

$$ex(m,n,C_{2t})=(t-1)n+m-t+1.$$

$$e(G-C)+e(G-C,C)\le \frac{(n-c)c}{2}.$$

$${\rho }_G(S):=e(G[S])+e_G(S,V(G)\backslash S).$$

$$\rho(x_{1},y_{1})=\left\{\begin{array}[]{ll}d(x_{1})+d(y_{1}),&x_{1}y_{1}\notin E(G);\\ d(x_{1})+d(y_{1})-1,&x_{1}y_{1}\in E(G).\end{array}\right.$$

$$\varrho (a,b)=e(L_{a,b})-c^2.$$

$$\theta_{X}=\left\{\begin{array}[]{ll}1,&u_{0}\in X;\\ 0,&u_{0}\in Y.\end{array}\right.\mbox{ and~{}~{}~{} }\theta_{Y}=\left\{\begin{array}[]{ll}0,&u_{0}\in X;\\ 1,&u_{0}\in Y.\end{array}\right.$$

$$\rho (G-C)={\rho }_{G_2}(G_2-C)+1\le \varrho (a-{\theta }_X,b-{\theta }_Y)+1\le \varrho (a,b),$$

$$G_3:=(G-E(u_0,B-u_0))\cup \{u_{3}v:v\in V(B),u_{0}v\in E(G)\}.$$

$${\rho }_{D_3}(D_3-C)\le \varrho (|X_{D_3}|,|Y_{D_3}|)={\rho }_{D_4}(D_4-C),$$

$$e(B_3)\le e(L_{|X_{B_3}|,|Y_{B_3}|})=e(B_4).$$

$${\rho }_{G_3}(G_3-C)={\rho }_{D_3}(D_3-C)+e(B_3)\le {\rho }_{D_4}(D_4-C)+e(B_4)={\rho }_{G_4}(G_4-C).$$

$$\vartheta_{x}=\left\{\begin{array}[]{ll}1,&u_{4}\in X;\\ 0,&u_{4}\in Y.\end{array}\right.\mbox{ and }\vartheta_{y}=\left\{\begin{array}[]{ll}0,&u_{4}\in X;\\ 1,&u_{4}\in Y.\end{array}\right.$$

$$B_5=L_{|X_{B_4}|-{\vartheta }_x,|Y_{B_4}|-{\vartheta }_y},D_5=L_{|X_{D_4}|+{\vartheta }_x,|Y_{D_4}|+{\vartheta }_y}.$$

$$\rho(x,y)=\left\{\begin{array}[]{ll}d(x)+d(y),&xy\notin E(G);\\ d(x)+d(y)-1,&xy\in E(G).\end{array}\right.$$

$${\rho }_{G_1}(G_1-C)\le c(|X_{G_1}|-1-c)+|Y_{G_1}|=(c+1)|X_H|,$$

$${\rho }_{G_2}(G_2-C)\le (c+1)|X_{G-C-H}|.$$

$$\rho (G-C)=\rho (H)+\rho (G-C-H)={\rho }_{G_1}(G_1-C)+{\rho }_{G_2}(G_2-C)\le (c+1)|X_{G-C}|=\varrho (a,b).$$

$$2c\ge |V(P_1)|+|V(P_2)|+\ell (\overleftarrow{C}[u_1,u_2])+1\ge 2{\beta }_1+2{\beta }_2+2c+1-\ell (\overrightarrow{C}[u_1,u_2]).$$

$$\ell (C)=\sum _{i=1}^{\alpha }\ell (\overrightarrow{C}[u_i,u_{i+1}])\ge 2{\beta }_1+2{\beta }_2+1+2(\alpha -1)>2c,$$

$$2c\ge |V(D)|+\ell (\overleftarrow{C}[u_1,u_2])+1\ge \beta +1+2c+1-\ell (\overrightarrow{C}[u_1,u_2]).$$

$$\ell (C)=\sum _{i=1}^{\alpha }\ell (C[u_i,u_{i+1}])\ge 2(\beta +2)+2(\alpha -2)>2c,$$

$$\ell (C)=\sum _{i=1}^{\alpha }\ell (C[u_i,u_{i+1}])\ge 2(\beta +1)+2(\alpha -2)>2c,$$

$$2c\ge |V(P)|\ge \beta +1+\ell (\overleftarrow{C}[u_i,u_{i+1}])+1=\beta +2c+2-\ell (\overrightarrow{C}[u_i,u_{i+1}]).$$

$$\ell (C)=\sum _{i=1}^{\alpha }\ell (C[u_i,u_{i+1}])\ge 2(\beta +2)+2(\alpha -2)>2c,$$

Full text

Exact bipartite Turán numbers of large even cycles

Binlong Li    Bo Ning Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, P.R. China. Email: binlongli@nwpu.edu.cn. Partially supported by the NSFC grant (No. 11601429).Corresponding author. Center for Applied Mathematics, Tianjin University, Tianjin 300072, P.R. China. Email: bo.ning@tju.edu.cn. Partially supported by the NSFC grant (No. 11601379) and the Seed Foundation of Tianjin University (2018XRG-0025).

Abstract: Let the bipartite Turán number $ex(m,n,H)$ of a graph $H$ be the maximum number of edges in an $H$-free bipartite graph with two parts of sizes $m$ and $n$, respectively. In this paper, we prove that $ex(m,n,C_{2t})=(t-1)n+m-t+1$ for any positive integers $m,n,t$ with $n\geq m\geq t\geq\frac{m}{2}+1$. This confirms the rest of a conjecture of Györi [15] (in a stronger form), and improves the upper bound of $ex(m,n,C_{2t})$ obtained by Jiang and Ma [20] for this range. We also prove a tight edge condition for consecutive even cycles in bipartite graphs, which settles a conjecture in [1]. As a main tool, for a longest cycle $C$ in a bipartite graph, we obtain an estimate on the upper bound of the number of edges which are incident to at most one vertex in $C$. Our two results generalize or sharpen a classical theorem due to Jackson [18] in different ways.

Keywords: bipartite Turán number; even cycle; bipartite graph; Györi’s conjecture

1 Introduction

We only consider simple graphs, which are undirected and finite. The study of cycles in bipartite graphs has a rich history. There are many results in the literature which use that a bipartite graph with high degree has a long cycle, see references [25, 6, 17, 18, 19]. Moreover, [21, 22] reveal that results on cycles in bipartite graphs play important roles in investigating cycles in hypergraphs. For a lot of references from the view of extremal graph theory, we refer to the survey [13].

Maybe one of the best known extremal results involving long cycles in bipartite graphs is the following proved more than 30 years old.

Theorem 1.1** (Jackson [18]).**

Let $t$ be an integer and $G=(X,Y;E)$ be a bipartite graph. Suppose that $|X|=n$, $|Y|=m$, where $n\geq m\geq t\geq 2$. Suppose that

[TABLE]

Then $G$ contains a cycle of length at least $2t$.

One question naturally arises: Can we find exact edge number conditions for cycles of given lengths? As we shall see later, we indeed have the following significant strengthening of Jackson’s theorem.

Theorem 1.2**.**

Let $t$ be an integer and $G=(X,Y;E)$ be a bipartite graph with $|X|=m$, $|Y|=n$. Suppose that $n\geq m$ and $t\leq m\leq 2t-2$. If $e(G)>(t-1)(n-1)+m$, then $G$ contains a cycle of length $2t$.

The above theorem in fact tells us exact information on “bipartite Tuán number” of large even cycles. Following Füredi [12], we define the bipartite Turán number $ex(m,n,H)$ of a graph $H$ to be the maximum number of edges in an $H$-free bipartite graph with two parts of sizes $m$ and $n$. In this paper, we mainly focus on the exact formula of $ex(m,n,C_{2t})$ for some range. For a similar problem on paths, Gyárfas, Rousseau, and Schelp [14] completely determined the function $ex(m,n,P_{t})$. When restricting to cycles, the situation turns out to be much more difficult.

Let us recall the classical result that $ex(n,n,C_{4})=(1+o(1))n^{\frac{3}{2}}$ due to Kővári, Sós, and Turán [23]. For the function $ex(m,n,C_{6})$, it is closely related to a number-theoretical problem on product representations of squares, which was studied by Erdős, Sárközy, and Sós in [10]. They conjectured that: (a) $ex(m,n,C_{6})<c(mn)^{\frac{2}{3}}$ when $n>m\geq n^{\frac{1}{2}}$; and (b) $ex(m,n,C_{6})<2n+c(mn)^{\frac{2}{3}}$ when $n\geq m^{2}$. The part (a) of this conjecture and a weaker result of part (b) were confirmed by Sárközy [27]. For the part (b), it was finally settled by Györi [15]. Interestingly, motivated by the extremal result on short cycles, Györi [15] suggested a general conjecture on longer cycles.

Conjecture 1** (Györi [15, p.373], see also [3]).**

Suppose that $m,n,k$ are integers, where $n\geq m^{2}$, $m\geq t\geq 3$. Then

[TABLE]

Using estimate on total weights of triangle-free multi-hypergraphs, Györi himself [16] disproved Conjecture 1 for the case $t=3$. Balbuena, García-Vázquez, Marcote, and Valenzuela [3] further disproved it when $t\leq\frac{m+1}{2}$. As far as we know, this conjecture remains open when $t\geq\frac{m}{2}+1$. The following conjecture sharpens the left part of Györi’s conjecture.

Conjecture 2**.**

Suppose that $m,n,t$ are positive integers, where $n\geq m\geq t\geq\frac{m}{2}+1$. Then

[TABLE]

For general results on $ex(m,n,C_{2t})$, an upper bound was obtained by Naor and Verstraëte [26], who proved that for $m\leq n$ and $t\geq 2$,

[TABLE]

Györi [15] proved that there exists some $c_{t}>0$ such that for $n\geq m^{2}$,

[TABLE]

Very recently, Jiang and Ma [20] proved the following new bound:

Theorem 1.3** (Jiang and Ma, Proposition 5.5 in [20]).**

There exists a constant $d_{t}>0$ such that for any positive integers $n\geq m\geq 2$,

[TABLE]

So, if Conjecture 2 is true, then it improves Jiang and Ma’s result for the range $t\geq\frac{m}{2}+1$.

In this paper, we aim to solve the aforementioned conjectures. In fact, we prove the following stronger result, which also confirms a conjecture in [1] (see Conjecture 1 in [1, p.30]).

Theorem 1.4**.**

Let $G=(X,Y;E)$ be a bipartite graph with $|X|=m$ and $|Y|=n$. Suppose that $n\geq m\geq 2k+2$ for some $k\in\mathbb{N}$. If $e(G)\geq n(m-k-1)+k+2$, then $G$ contains cycles of all even lengths from 4 up to $2m-2k$.

Set $t=m-k$ in Conjecture 2. Let $G$ be a graph obtained by identifying one vertex in the $n$-set from $K_{m-k-1,n}$ and the other vertex in the $1$-set from $K_{1,k+1}$. Then a longest cycle in $G$ is of length $2m-2k-2$ and $e(G)=(m-k-1)n+k+1$. This tells us $ex(m,n,C_{2t})\geq(m-k-1)n+k+1$. An immediate consequence of Theorem 1.4 is that $ex(m,n,C_{2t})\leq(m-k-1)n+k+1$. Thus, we have the following result, which is equivalent to Theorem 1.2 and confirms Conjecture 2.

Corollary 1**.**

For any positive integers $m,n,t$, if $n\geq m\geq t\geq\frac{m}{2}+1$, then

[TABLE]

The basic case of the proof of Theorem 1.4 is that the bipartite graph is balanced. For this special case, a slightly stronger theorem will be proved.

Theorem 1.5**.**

*Let $G$ be a balanced bipartite graph of order $2n$, where $n\geq 2k+2$, $k\in\mathbb{N}$. If $e(G)\geq(n-k-1)n+k+2$, then there hold:

(i) The circumference $c(G)\geq 2n-2k$; and (ii) $G$ contains cycles of all even lengths from 4 to $c(G)$.*

Our proof of Theorem 1.5 is motivated by a theorem of Bondy [5] stated as follows, which extends the celebrated Erdős-Gallai Theorem [9] on cycles.

Theorem 1.6** (Bondy [5]).**

Let $G$ be a graph on $n$ vertices and $C$ a longest cycle of $G$ with order $c$. Then

[TABLE]

Since $G[C]$ contains at most $\binom{c}{2}$ edges, it can imply Erdős-Gallai Theorem. In fact, Bondy’s theorem and its variants turned out to be powerful tools for tacking many problems on long cycles. For example, it actually plays an important role in Bollobás and Thomason’s almost proof [4] of Brandt’s conjecture [7], which says that every non-bipartite graph on $n$ vertices is weakly pancyclic if $e(G)\geq\lfloor\frac{n^{2}}{4}\rfloor-n+5$. The other example is that, Bondy’s theorem is related to a conjecture of Woodall [28] in 1976. Ma and one of authors here [24] recently proved a stability version of Bondy’s theorem, which is one step towards obtaining a stability version of Woodall’s conjecture [28].

Very importantly for us, using Theorem 1.6 is an ingenious idea in Bondy’s proof of Turán numbers of large cycles [5]. We shall prove a bipartite analog of Theorem 1.6 and use it to prove Theorem 1.5.

Theorem 1.7**.**

*Let $G=(X,Y;E)$ be a bipartite graph and $C$ a longest cycle of $G$. Suppose that $|X|=n$, $|Y|=m$, and $|V(C)|=2t$, where $n\geq m\geq t$. Then there hold:

(1) If $m\leq 2t$, then $e(G-C)+e(G-C,C)\leq t(n-1-t)+m$.

(2) If $m\geq 2t$, then $e(G-C)+e(G-C,C)\leq t(m+n+1-3t)$.*

The bounds in Theorem 1.7 are tight. We postpone the discussion to Section 2. Moreover, Theorem 1.7 generalizes Theorem 1.1 in the other direction.

Let us digest some notation and terminologies. Let $G=(X,Y;E)$ be a bipartite graph, where $X,Y$ are two bipartite sets and $E$ is the edge set of $G$. We say that $G$ is balanced if $|X|=|Y|$. Let $P$ be a path of $G$. We say that $P$ is an $(x,y)$-path if $x,y$ are two end-vertices of $P$; and $P$ is an $x$-path if $x$ is one end-vertex of $P$. A graph $G$ is called weakly pancyclic if $G$ contains all cycles of lengths from $g(G)$ to $c(G)$, where $g(G)$ and $c(G)$ are its girth and circumference, respectively. A balanced bipartite graph $G$ is called bipancyclic, if $G$ contains all cycles of even lengths from 4 to $2|X|$. For a subgraph $H$ of $G$, we set $X_{H}=X\cap V(H)$ and $Y_{H}=Y\cap V(H)$. We use $|H|$ to denote the order of $H$, that is, $|H|:=|V(H)|$. Let $S\subseteq V(G)$. We use $G[S]$ to denote the subgraph induced by $S$, and $G-S$ the subgraph induced by $V(G-S)$. Specially, when there is no danger of ambiguity, we use $G-H$ instead of $G-V(H)$ sometimes. For $V_{1},V_{2}\subseteq V(G)$ with $V_{1}\cap V_{2}=\emptyset$, we set $e(V_{1},V_{2})=\{v_{1}v_{2}:v_{1}\in V_{1},v_{2}\in V_{2}\}$. If $H_{1},H_{2}$ are two disjoint subgraphs of $G$, then we set $e(H_{1},H_{2})=e(V(H_{1}),V(H_{2}))$.

The remainder of this paper is organized as follows. In Section 2, we aim to prove Theorem 1.7. In Subsection 2.1, we prove several technical lemmas and list useful theorems. In Subsection 2.2, we prove Theorem 1.7. In Section 3, we prove Theorem 1.5 and Theorem 1.4.

2 Proof of Theorem 1.7

2.1 Preliminaries for proving Theorem 1.7

In this subsection, we collect and establish several lemmas to be used later. Let $G$ be a graph and $P$ be a path of $G$ with the origin $x$ and terminus $y$. The path $P$ is called a maximal path of $G$, if $N_{G}(x)\cup N_{G}(y)\subseteq V(P)$. We say that $P$ is a maximal $x$-path of $G$ if $N_{G}(y)\subseteq V(P)$.

Lemma 2.1**.**

*Let $G=(X,Y;E)$ be a connected bipartite graph and $d(x)\geq d$ for every vertex $x\in X$.

(1) If $|X|\geq|Y|$, then for every vertex $y_{0}\in Y$, $G$ has a maximal $y_{0}$-path with the terminus in $X$ and of order at least $2d$.

(2) If $|X|>|Y|$, then for every vertex $x_{0}\in X$, $G$ has a maximal $x_{0}$-path with the terminus in $X$ and of order at least $2d+1$.*

Proof.

We first show the existence of a maximal $x_{0}$- or $y_{0}$-path with the terminus in $X$. We use induction on $n:=|V(G)|$. The assertion is trivial if $n=1,2$. Suppose that $n\geq 3$.

First assume $|X|\geq|Y|$ and let $y_{0}\in Y$. Thus, $|X|>|Y\backslash\{y_{0}\}|$. It follows that there is a component $H$ of $G-y_{0}$ (possibly $H=G-y_{0}$) such that $|X_{H}|>|Y_{H}|$. Since $G$ is connected, $y_{0}$ has a neighbor $x_{0}\in X_{H}$. By the induction hypothesis, $H$ has a maximal $x_{0}$-path $P_{0}$ with the terminus in $X_{H}\subseteq X$. Thus, $P=y_{0}x_{0}P_{0}$ is a maximal $y_{0}$-path with the terminus in $X$.

Now assume $|X|>|Y|$ and let $x_{0}\in X$. Thus, $|X\backslash\{x_{0}\}|\geq|Y|$. It follows that there is a component $H$ of $G-x_{0}$ (possibly $H=G-x_{0}$) such that $|X_{H}|\geq|Y_{H}|$. Since $G$ is connected, $x_{0}$ has a neighbor $y_{0}\in Y_{H}$. By the induction hypothesis, $H$ has a maximal $y_{0}$-path $P_{0}$ with the terminus in $X_{H}\subseteq X$. Thus, $P=x_{0}y_{0}P_{0}$ is a maximal $x_{0}$-path with the terminus in $X$.

It remains to show that the path $P$ is of order at least $2d$ (if $P$ originates at $y_{0}$), or at least $2d+1$ (if $P$ originates at $x_{0}$). Let $x_{1}\in X$ be the terminus of $X$ other than $x_{0}$ or $y_{0}$. Notice that $d(x_{1})\geq d$ and $N(x_{1})\subseteq V(P)$. We have $|Y\cap V(P)|\geq d$, implying that $P$ has order at least $2d$ when $P$ originates at $y_{0}$, and at least $2d+1$ when $P$ originates at $x_{0}$. This proves Lemma 2.1.

For a graph $G$ and $S\subseteq V(G)$, we denote by $\rho_{G}(S)$ the number of edges in $G$ which are incident to at least one vertex in $S$, that is,

[TABLE]

From this definition, one can see $d(v)=\rho_{G}(\{v\})$ for any vertex $v\in V(G)$. When there is no danger of ambiguity, we use $\rho(u,v)$ and $\rho(S)$ instead of $\rho_{G}(\{u,v\})$ and $\rho_{G}(S)$, respectively. An $\{s,s^{\prime}\}$-disjoint path pair of $G$ (or shortly, an $\{s,s^{\prime}\}$-DPP), is the union of an $s$-path and an $s^{\prime}$-path which are vertex-disjoint. Let $D$ be an $\{s,s^{\prime}\}$-DPP, and $t,t^{\prime}$ be the termini of the two paths in $D$. We say that $D$ is a maximal $\{s,s^{\prime}\}$-DPP in $G$ if $N_{G}(t)\cup N_{G}(t^{\prime})\subseteq V(D)$. Clearly, $D$ is a maximal $\{s,s^{\prime}\}$-DPP of $G$, if and only if $D+ss^{\prime}$ is a maximal path of $G+ss^{\prime}$. For a special case that $G$ is bipartite, we say that $D$ is detached if $t$ and $t^{\prime}$ are in distinct partition sets of $G$.

Next we shall prove two lemmas on degree conditions for detached maximal DDP in bipartite graphs.

Lemma 2.2**.**

Let $G=(X,Y;E)$ be a connected balanced bipartite graph. If $\rho(x,y)\geq\rho$ for every $(x,y)\in(X,Y)$, then for any $(x_{0},y_{0})\in(X,Y)$, $G$ has a detached maximal $\{x_{0},y_{0}\}$-DPP of order at least $\rho+1$.

Proof.

We first show the existence of the detached maximal $\{x_{0},y_{0}\}$-DPP by induction on $n:=|V(G)|$. The assertion is trivial if $n=2$. So assume that $n\geq 4$. Let $x_{0}\in X,y_{0}\in Y$ and let $G^{\prime}:=G-\{x_{0},y_{0}\}$.

First assume that there is a balanced component $H$ of $G^{\prime}$ that is incident to both $x_{0}$ and $y_{0}$. Let $x_{1},y_{1}\in V(H)$ be the neighbors of $y_{0}$ and $x_{0}$, respectively. By the induction hypothesis, $H$ has a detached maximal $\{x_{1},y_{1}\}$-DPP, say $D_{0}$. Thus, $D=D_{0}\cup\{x_{0}y_{1},x_{1}y_{0}\}$ is a detached maximal $\{x_{0},y_{0}\}$-DPP of $G$.

Now assume that every balanced component of $G^{\prime}$ is incident to either $x_{0}$ or $y_{0}$ but not both. Let $\mathcal{H}_{1}$ be the set of components $H$ of $G^{\prime}$ such that either $|X_{H}|>|Y_{H}|$, or $H$ is balanced and incident to $x_{0}$. Let $\mathcal{H}_{2}$ be the set of components $H$ of $G^{\prime}$ such that either $|Y_{H}|>|X_{H}|$, or $H$ is balanced and incident to $y_{0}$.

If $y_{0}$ is not incident to any component of $G^{\prime}$, then every component of $G^{\prime}$ is incident to $x_{0}$. This fact implies that $\mathcal{H}_{1}\neq\emptyset$. Let $H\in\mathcal{H}_{1}$ and $y_{1}\in N_{H}(x_{0})$. By Lemma 2.1(1), $H$ has a maximal $y_{1}$-path $P_{1}$ with the terminus in $X_{H}\subseteq X$. Thus, the union of the path $P_{x}=x_{0}y_{1}P_{1}$ and the trivial path $P_{y}=y_{0}$ is a detached maximal $\{x_{0},y_{0}\}$-DPP of $G$, and we are done.

In the following, we assume $y_{0}$ is incident to at least one component of $G^{\prime}$; and similarly, by symmetry, $x_{0}$ is incident to at least one component of $G^{\prime}$.

If $\mathcal{H}_{2}=\emptyset$, then every component of $G^{\prime}$ is balanced, and it follows that $y_{0}$ is not incident to any component of $G^{\prime}$, a contradiction. So $\mathcal{H}_{2}\neq\emptyset$, and similarly, $\mathcal{H}_{1}\neq\emptyset$.

It follows that there exist $H_{1}\in\mathcal{H}_{1}$ and $H_{2}\in\mathcal{H}_{2}$ such that: $x_{0}$ is incident to one of $H_{1}$ and $H_{2}$, and $y_{0}$ is incident to the other. If $x_{0}$ is incident to $H_{1}$ and $y_{0}$ is incident to $H_{2}$, then let $y_{1}\in N_{H_{1}}(x_{0})$ and $x_{1}\in N_{H_{2}}(y_{0})$. Recall that $|X_{H_{1}}|\geq|Y_{H_{1}}|$ and $|Y_{H_{2}}|\geq|X_{H_{2}}|$. By Lemma 2.1, $H_{1}$ has a maximal $y_{1}$-path $P_{1}$ with the terminus in $X_{H_{1}}\subseteq X$ and $H_{2}$ has a maximal $x_{1}$-path $P_{2}$ with the terminus in $Y_{H_{2}}\subseteq Y$. Thus, the union of the two paths $P_{x}=x_{0}y_{1}P_{1}$ and $P_{y}=y_{0}x_{1}P_{2}$ is a detached maximal $\{x_{0},y_{0}\}$-DPP of $G$. If $x_{0}$ is incident to $H_{2}$ and $y_{0}$ is incident to $H_{1}$, then let $y_{1}\in N_{H_{2}}(x_{0})$ and $x_{1}\in N_{H_{1}}(y_{0})$. Note that in this case $|X_{H_{1}}|>|Y_{H_{1}}|$ and $|Y_{H_{2}}|>|X_{H_{2}}|$. By Lemma 2.1, $H_{2}$ has a maximal $y_{1}$-path $P_{1}$ with the terminus in $Y_{H_{2}}\subseteq Y$, and $H_{1}$ has a maximal $y_{1}$-path $P_{2}$ with the terminus in $X_{H_{1}}\subseteq X$. Thus, the union of the two paths $P_{x}=x_{0}y_{1}P_{1}$ and $P_{y}=y_{0}x_{1}P_{2}$ is a detached maximal $\{x_{0},y_{0}\}$-DPP of $G$. This proves the existence of the detached maximal $\{x_{0},y_{0}\}$-DPP of $G$.

Now let $D$ be a detached maximal $\{x_{0},y_{0}\}$-DPP of $G$. We will show that $D$ has order at least $\rho+1$. Let $x_{1}\in X,y_{1}\in Y$ be the termini of the two paths in $D$. Obviously, we have

[TABLE]

If $x_{1}y_{1}\notin E(G)$, then $|V(D)|\geq d(x_{1})+d(y_{1})+2\geq\rho+1$; if $x_{1}y_{1}\in E(G)$, then $|V(D)|\geq d(x_{1})+d(y_{1})\geq\rho+1$. This proves Lemma 2.2.

Let $G$ be a graph with connectivity 1, and $u,v\in V(G)$. We call $\{u,v\}$ a good pair of $G$, if there is an end-block $B$ of $G$ such that exactly one of $u,v$ is an inner-vertex of $B$.

Lemma 2.3**.**

Let $G=(X,Y;E)$ be a balanced bipartite graph with connectivity 1, and $\{x_{0},x^{\prime}_{0}\}\subseteq X$ be a good pair of $G$. Suppose $\rho(x,y)\geq\rho$ for every $(x,y)\in(X,Y)$. Then $G$ has a detached maximal $\{x_{0},x^{\prime}_{0}\}$-DPP of order at least $\rho+1$.

Proof.

We prove the assertion by induction on $|V(G)|$. It is easy to check that the assertion is true for $n=4$. Now assume that $n\geq 6$. If $G$ has a detached maximal $\{x_{0},x^{\prime}_{0}\}$-DPP, say $D$, then $D$ has order at least $\rho+1$ (see the last paragraph of the proof of Lemma 2.2), and the statement holds. Now we assume that $G$ has no detached maximal $\{x_{0},x^{\prime}_{0}\}$-DPP. Let $B$ be an end-block of $G$ that contains one vertex, say $x^{\prime}_{0}$, as an inner-vertex, and let $u$ be the cut-vertex of $G$ contained in $B$. If $x_{0}$ is also an inner-vertex of an end-blocks, say $B_{0}$, then we assume without loss of generality that $|V(B)|\leq|V(B_{0})|$.

Suppose first that $x_{0}$ is a cut-vertex of $G$. Let $H$ be the component of $G-x_{0}$ containing $x^{\prime}_{0}$.

If $H$ is balanced, then let $y_{0}$ be a neighbor of $x_{0}$ in $H$. By Lemma 2.2, $H$ contains an detached maximal $(x^{\prime}_{0},y_{0})$-DPP, say $D$. It follows that $D\cup\{x_{0}y_{0}\}$ is an detached maximal $(x_{0},x^{\prime}_{0})$-DPP of $G$, a contradiction. So we assume that $H$ is not balanced.

If $|X_{H}|>|Y_{H}|$, then $|X_{G-H}|<|Y_{G-H}|$. By Lemma 2.1, $H$ has a maximal $x^{\prime}_{0}$-path with terminus in $X_{H}$ and $G-H$ has a maximal $x_{0}$-path with terminus in $Y_{G-H}$. Thus the union of such two paths form an detached maximal $(x_{0},x^{\prime}_{0})$-DPP of $G$, a contradiction. If $|X_{H}|<|Y_{H}|$, then $|X_{G-H}|>|Y_{G-H}|$. By Lemma 2.1, $H$ has a maximal $x^{\prime}_{0}$-path with terminus in $Y_{H}$ and $G-H$ has a maximal $x_{0}$-path with terminus in $X_{G-H}$. Thus the union of such two paths form a detached maximal $(x_{0},x^{\prime}_{0})$-DPP of $G$, also a contradiction.

Now we assume that $G^{\prime}=G-x_{0}$ is connected. Specially we have $x_{0}\in V(G-B)$. Here we deal with the case that $N(x_{0})=\{u\}$. For this case $x_{0}$ is an inner-vertex of the end-block $B_{0}$ with $V(B_{0})=\{x_{0},u\}$. It follows that $V(B)=\{x^{\prime}_{0},u\}$. By Lemma 2.1, $G-\{x_{0},x^{\prime}_{0}\}$ has a maximal $u$-path $P$ with terminus in $Y$. Now the two paths $P_{1}=x_{0}uP$ and $P_{2}=x^{\prime}_{0}$ form a detached maximal $(x_{0},x^{\prime}_{0})$-DPP of $G$, a contradiction. So we assume that $x_{0}$ has a neighbor $y_{0}\in V(G-B)$. We choose $y_{0}$ such that the distance between $y_{0}$ and $u$ in $G$ is as large as possible. It follows that $B$ is an end-block of $G^{\prime}$ as well.

Let $H$ be the component of $G^{\prime}-y_{0}$ containing $x^{\prime}_{0}$. So $B$ is contained in $H$.

We claim that $B$ is an end-block of $H$ as well. Suppose not. Then $B=H$. This implies that $N_{H}(y_{0})=\{u\}$, specially $u\in X$. If $x_{0}$ has a second neighbor $y_{1}$, then the distance between $u$ and $y_{1}$ is larger than that between $u$ and $y_{0}$, a contradiction. It follows that $N(x_{0})=\{y_{0}\}$. Now $x_{0}$ is an inner-vertex of the end-block $B_{0}$ with $V(B_{0})=\{x_{0},y_{0}\}$, which contradicting our choice of $B$. Thus as we claimed, $B$ is an end-block of $H$ as well.

If $H$ is balanced, then let $x_{1}=N_{H}(y_{0})$. Since $y_{0}\in V(G-B)$, we have that $\{x_{1},x^{\prime}_{0}\}$ is a good pair of $H$. By induction, $H$ has a detached maximal $(x_{1},x^{\prime}_{0})$-DPP $D$. Now $D\cup\{x_{0}y_{0},y_{0}x_{1}\}$ is a detached maximal $(x_{1},x^{\prime}_{0})$-DPP of $G$, a contradiction. So we assume that $H$ is not balanced.

If $|X_{H}|>|Y_{H}|$, then $|X_{G-H}|<|Y_{G-H}|$. By Lemma 2.1, $H$ has a maximal $x^{\prime}_{0}$-path $P_{1}$ with terminus in $X_{H}$ and $G-H$ has a maximal $y_{0}$-path $P_{2}$ with terminus in $Y_{G-H}$. If $|X_{H}|<|Y_{H}|$, then $|X_{G-H}|\geq|Y_{G-H}|$. By Lemma 2.1, $H$ has a maximal $x^{\prime}_{0}$-path $P_{1}$ with terminus in $Y_{H}$ and $G-H$ has a maximal $y_{0}$-path with terminus in $X_{G-H}$. In both cases, the two paths $P_{1}$ and $x_{0}y_{0}P_{2}$ form a detached maximal $(x_{0},x^{\prime}_{0})$-DPP of $G$, a contradiction.

Lemma 2.4** (Jackson [18]).**

Let $G=(X,Y,E)$ be a 2-connected bipartite graph, where $|X|\geq|Y|$. If each vertex of $X$ has degree at least $k$, and each vertex of $Y$ has degree at least $l$, then $G$ contains a cycle of length at least $2\min\{|Y|,k+l-1,2k-2\}$. Moreover, if $k=l$ and $|X|=|Y|$, then $G$ contains a cycle of length at least $2\min\{|Y|,2k-1\}$.

Lemma 2.5** (Bagga, Varma [2]).**

Let $G=(X,Y,E)$ be a balanced bipartite graph of order $2n$. If $d(x)+d(y)\geq n+2$ for every $(x,y)\in(X,Y)$, then $G$ is Hamilton-biconnected.

Lemma 2.6**.**

Let $G$ be a 2-connected balanced bipartite graph such that $\rho(x,y)\geq\rho$ for every $(x,y)\in(X,Y)$. Then for any two vertices $x_{1},x_{2}\in X$, $G$ has an $(x_{1},x_{2})$-path of order at least $\rho$.

Proof.

The assertion can be checked easily when $|X|=2$. So we assume that $|X|\geq 3$.

Set $k=\min\{d(x):x\in X\}$ and $l=\min\{d(y):y\in Y\}$. It follows that $k+l=d(x)+d(y)\geq\rho(x,y)\geq\rho$ for some $(x,y)\in(X,Y)$. If $k\neq l$, then $2k-2\geq\rho-1$ or $2l-2\geq\rho-1$; if $k=l$, then we have $2k-1\geq\rho-1$. Notice that $|X|=|Y|$. By Lemma 2.4, $G$ has a cycle of length at least $2\min\{|X|,\rho-1\}$.

Suppose first that $|X|\geq\rho-1$. It follows that $G$ has a cycle $C$ of length at least $2\rho-2$. Since $G$ is 2-connected, there are two disjoint paths $P_{1},P_{2}$ between $x_{1},x_{2}$, respectively, and $C$. Let $u_{1},u_{2}\in V(C)$ be the terminus of $P_{1},P_{2}$ (possibly $x_{i}=u_{i}$ for some $i=1,2$). Now one of the two paths $\overrightarrow{C}[u_{1},u_{2}]$ and $\overleftarrow{C}[u_{1},u_{2}]$ has order at least $\rho$. Together with $P_{1},P_{2}$, we can get an $(x_{1},x_{2})$-path of order at least $\rho$.

Secondly, we suppose that $|X|=\rho-2$. It follows that $G$ has a Hamiltonian cycle $C$. If one of the two paths $P_{1}=\overrightarrow{C}[x_{1},x_{2}]$ and $P_{2}=\overleftarrow{C}[x_{1},x_{2}]$ has order at least $\rho$, then there are noting to prove. So we assume that both $P_{1}$ and $P_{2}$ has order exactly $\rho-1$. If there is an edge, say $u_{1}u_{2}$, with $u_{i}\in V(P_{i})\backslash\{x_{1},x_{2}\}$, then one of the paths $x_{1}P_{1}u_{1}u_{2}P_{2}x_{2}$ and $x_{1}P_{2}u_{2}u_{1}P_{1}x_{2}$ has order at least $\rho$ (notice that the sum of the orders of such two paths is $2\rho$), and we are done. Now we assume that there are no edges between $V(P_{1})\backslash\{x_{1},x_{2}\}$ and $V(P_{2})\backslash\{x_{1},x_{2}\}$. It follows that for any two vertices $(x,y)\in(X\backslash\{x_{1},x_{2}\},Y)$, $d(x)+d(y)\leq\rho-1$, a contradiction.

Lastly, we suppose that $|X|\leq\rho-3$. Let $y_{1}\in N(x_{1})$, $y_{2}\in Y\backslash\{y_{1}\}$, and set $G^{\prime}=G-\{x_{1},y_{2}\}$. Now $G^{\prime}$ is a balanced bipartite graph of order $2(\rho-4)$ and for every $(x,y)\in(X_{G^{\prime}},Y_{G^{\prime}})$, $d_{G^{\prime}}(x)+d_{G^{\prime}}(y)\geq\rho_{G^{\prime}}(x,y)\geq\rho-2$. By Lemma 2.5, $G^{\prime}$ is Hamilton-biconnected. Let $P^{\prime}$ be a Hamiltonian $(y_{1},x_{2})$-path of $G^{\prime}$. Then $P=x_{1}y_{1}P^{\prime}$ is an $(x_{1},x_{2})$-path of order $|V(G)|-1$. Notice that if $G$ is complete and bipartite, then $\rho\leq|V(G)|-1$; otherwise $\rho\leq|V(G)|-2$. It follows that $P$ is an $(x_{1},x_{2})$-path of order at least $\rho$.

A subgraph $F$ of a graph $G$ is called an $(x,L)$-fan if $F$ has the following decomposition $F=\cup_{i=1}^{k}P_{i}$, where

• •

$k\geq 2$;

• •

each $P_{i}$ is a path with two end-vertices $x$ and $y_{i}\in V(L)$;

• •

$V(P_{i})\cap V(L)=\{y_{i}\}$ and $V(P_{i})\cap V(P_{j})=\{x\}$.

The following lemma on the fan structure is a corollary of a theorem on weighted graphs, which was proved by Fujisawa, Yoshimoto, and Zhang (see [11, Lemma 1]). We need this refined version of the Fan Lemma to find a long cycle.

Lemma 2.7**.**

Let $G$ be a 2-connected graph, $C$ a longest cycle $G$, and $H$ a component of $G-C$. If $d(v)\geq d$ for every $v\in V(H)$, then for every vertex $x\in V(H)$, there is an $(x,C)$-fan $F$ with $e(F)\geq d$.

We also need the following two results on long cycles in bipartite graphs due to Jackson [17, 18].

Lemma 2.8**.**

(Jackson [18, Lemma 5]) Let $G=(X,Y;E)$ be a 2-connected bipartite graph. Let $P$ be a maximal path in $G$ with two end-vertices $u$ and $v$.

(i) If $u\in X$ and $v\in Y$, then $G$ contains a cycle of length at least $\min\{|V(P)|,2(d(u)+d(v)-1)\}$.

(ii) If $u,v\in X$ then $G$ contains a cycle of length at least $\min\{|V(P)|-1,2(d(u)+d(v)-2)\}$.

The final lemma was originally conjectured by Sheehan (see [17, pp.332]).

Lemma 2.9**.**

(Jackson [17, Theorem 1])* Let $G=(X,Y;E)$ be a bipartite graph with $|X|\leq|Y|$. If $d(x)\geq\max\{|X|,|Y|/2+1\}$ for every vertex $x\in X$, then $G$ has a cycle containing all vertices in $X$.*

2.2 Proof of Theorem 1.7

We first discuss on the extremal graphs of Theorem 1.7. Before the discussion, let us introduce some notations.

Let $a$, $b$, and $c$ be three positive integers. Let $\mathcal{B}_{a,b}$ be the set of bipartite graphs with two partition sets of size $a$ and $b$, respectively. We define a graph $L_{a,b}^{c}$ as follows. (If the notation $c$ is unemphatic, we use $L_{a,b}$ instead.)

$\bullet$ If $a\leq c$ or $b\leq c$, then let $L_{a,b}=K_{a,b}$.

$\bullet$ If $c<b\leq\max\{a,2c\}$, then let $L_{a,b}$ be the graph by identifying one vertex from the $a$-set of $K_{a,c}$ and the other one from the $1$-set of $K_{1,b-c}$.

$\bullet$ If $c<a\leq\max\{b,2c\}$, then let $L_{a,b}$ be the graph by identifying one vertex from the $b$-set of $K_{b,c}$ and the other vertex from the $1$-set of $K_{1,a-c}$.

$\bullet$ If $2c<\max\{a,b\}$, then let $L_{a,b}$ be the graph by identifying one vertex from the $c$-set of $K_{c,b-c}$ and the other vertex from the $(a-c+1)$-set of $K_{a-c+1,c}$.

By Theorem 1.1, $L_{a,b}^{c}$ is a graph in $\mathcal{B}_{a,b}$ with the maximum number of edges among those without cycles of length more than $2c$ (see Jackson [18]). One can see the graph $L_{a,b}^{c}$ shows that the bounds in Theorem 1.7 are tight for each case.

We define

[TABLE]

Notice that if $C$ is a longest cycle of a graph $G=L_{a,b}^{c}$, $c\leq b\leq a$, then $\varrho(a,b)=\rho(G-C)$.

Armed with the necessary additional notations, let us restate Theorem 1.7 as follows.

Theorem $1.7^{{}^{\prime}}$. Let $G=(X,Y;E)$ be a bipartite graph, where $|X|=a\geq b=|Y|$. Let $C$ be a longest cycle of $G$ of length $2c$. Then $\rho(G-C)\leq\varrho(a,b)$, i.e.,

(1) if $b\leq 2c$, then $\rho(G-C)\leq c(a-1-c)+b$; and

(2) if $b\geq 2c$, then $\rho(G-C)\leq c(a+b+1-3c)$.

Now we give a proof of Theorem $1.7^{{}^{\prime}}$.

Proof of Theorem $1.7^{{}^{\prime}}$. We prove the theorem by contradiction. Let $G$ be a counterexample to Theorem $1.7^{{}^{\prime}}$ such that:

(i) $|G|$ is minimum; and

(ii) subject to (i), $e(G)$ is maximum.

Claim 1**.**

$G$ is 2-connected.

Proof.

If $G$ is disconnected, then let $G_{1}$ be a connected bipartite graph obtained from $G$ by adding $\omega(G)-1$ edges such that each new edge is between distinct partition sets, where $\omega(G)$ denotes the number of components in $G$. Since the added edges are cut-edges of $G_{1}$, $C$ is a longest cycle of $G_{1}$ as well and all the add edges are outside $C$. So $\rho_{G}(G-C)<\rho_{G_{1}}(G_{1}-C)$, but $e(G_{1})>e(G)$, a contradiction to (ii). This implies $G$ is connected. Suppose now that $G$ has connectivity 1. Let $B$ be an end-block of $G$ with smallest order among those not containing $C$. We choose $G$ such that:

(iii) subject to (i),(ii), $|B|$ is minimum.

Let $u_{0}$ the cut-vertex of $G$ contained in $B$. Set

[TABLE]

We first claim that $|X_{B}|\geq 2$ and $|Y_{B}|\geq 2$. Suppose not. Since $B$ is bipartite and non-separable, we deduce $B\cong K_{2}$. Set $V(B)=\{u_{0},v_{0}\}$. So $v_{0}$ is of degree 1. Let $G_{2}:=G-v_{0}$. By the choice of $G$, we get

[TABLE]

a contradiction. Thus, $|X_{B}|\geq 2$ and $|Y_{B}|\geq 2$.

Let $u_{3}\in V(C)$ such that $u,u_{3}$ are in the same partition set (possibly $u_{0}=u_{3}$). Let

[TABLE]

Clearly, $C$ is a longest cycle of $G_{3}$ as well, and $\rho(G-C)=\rho_{G_{3}}(G_{3}-C)$. Hence $G_{3}$ is also a counterexample satisfying (i)(ii)(iii).

We use $B_{3}$ to denote the end-block of $G_{3}$ with the vertex set $(V(B)\backslash\{u_{0}\})\cup\{u_{3}\}$. Set $D_{3}=G_{3}-(B_{3}-u_{3})$. So $G_{3}$ consists of $B_{3}$ and $D_{3}$. We have $|X_{B_{3}}|+|X_{D_{3}}|=a+\theta_{X}$, and $|Y_{B_{3}}|+|Y_{D_{3}}|=b+\theta_{Y}$. Since $D_{3}$ contains the cycle $C$, $|X_{D_{3}}|\geq c$ and $|Y_{D_{3}}|\geq c$.

Let $B_{4}$ be a graph on $V(B_{3})$ isomorphic to $L_{|X_{B_{3}}|,|Y_{B_{3}}|}$ (recall the definition), $D_{4}$ a graph on $V(D_{3})$ isomorphic to $L_{|X_{D_{3}}|,|Y_{D_{3}}|}$, and $G_{4}$ the union of $B_{4}$ and $D_{4}$. Clearly, the longest cycle of $D_{4}$ is of length $2c$. We choose $D_{4}$ such that $C$ is a (longest) cycle in $D_{4}$ as well. Also note that $B_{3}$ has no cycle of length more than $2c$. By the choice of $G$,

[TABLE]

and furthermore, by Theorem 1.1,

[TABLE]

Thus, we have

[TABLE]

Since $e(G_{4}[C])=c^{2}$ and $e(G_{4})\geq e(G_{3})$, we can see that $G_{4}$ is a counterexample satisfying (i)(ii)(iii).

If $B_{4}$ is separable, then we have a contradiction to (iii). So $B_{4}$ is 2-connected, i.e., $B_{4}$ is an end-block of $G_{4}$. By the definition of $B_{4}\cong L_{|X_{B_{3}}|,|Y_{B_{3}}|}$, we infer $\min\{|X_{B_{4}}|,|Y_{B_{4}}|\}\leq c$ and $B_{4}\cong K_{|X_{B_{3}}|,|Y_{B_{3}}|}$. Recall that $u_{3}$ is the cut-vertex of $G_{4}$ contained in $B_{4}$. Let $u_{4}$ be a vertex in $V(B_{4})\backslash\{u_{3}\}$ with $d_{B_{4}}(u_{4})\leq c$ (recall that $|X_{B_{4}}|\geq 2$ and $|Y_{B_{4}}|\geq 2$). Set

[TABLE]

We will show that $\min\{|X_{D_{4}}|,|Y_{D_{4}}|\}=c$. Suppose that $\min\{|X_{D_{4}}|,|Y_{D_{4}}|\}>c$. Without loss of generality, we assume $|X_{D_{4}}|\geq|Y_{D_{4}}|$ (the other case can be dealt with similarly). If $c<|Y_{D_{4}}|\leq 2c$, then $D_{4}$ has a pendent edge, a contradiction to (iii). So $|Y_{D_{4}}|>2c$, and $D_{4}$ consists of $A_{x}\cong K_{c,|Y_{D_{4}}|-c}$ and $A_{y}\cong K_{|X_{D_{4}}|+1-c,c}$, with a common vertex in $X_{D_{4}}$. Let $G_{5}$ be the graph obtained from $G_{4}-E(u_{4},B_{4})$ by adding edges from $u_{4}$ to all vertices in $X_{A_{x}}$ (if $u_{4}\in Y_{B_{4}}$) or $Y_{A_{y}}$ (if $u_{4}\in X_{B_{4}}$). In each case, the vertex $u_{4}$ is of degree $c$ in $G_{5}$. That is, $G_{5}=B_{5}\cup D_{5}$ where

[TABLE]

Obviously, $C$ is a longest cycle of $G_{5}$ as well, and $\rho_{G_{4}}(G_{4}-C)\leq\rho_{G_{5}}(G_{5}-C)$. Note that the end-block $B_{5}$ of $G_{5}$ has order less than $B$, a contradiction to (iii). Thus, $\min\{|X_{D_{4}}|,|Y_{D_{4}}|\}=c$.

By symmetry, we assume that $|X_{D_{4}}|=c$. If $|X_{B_{4}}|\leq c$, then we can get a contradiction similarly as above. So assume that $|X_{B_{4}}|>c$, which implies that $|Y_{B_{4}}|\leq c$, since $\min\{|X_{B_{4}}|,|Y_{B_{4}}|\}\leq c$.

We claim that $|Y_{B_{4}}|=c$. Suppose that $|Y_{B_{4}}|<c$. If $|Y_{D_{4}}|=c$ as well, then we can get a contradiction similarly as above. So $|Y_{D_{4}}|>c$. Let $u_{6}$ be a vertex in $Y_{D_{4}-C}$, and let $G_{6}$ be the graph obtained from $G_{4}-E(u_{6},D_{4})$ by adding edges from $u_{6}$ to all vertices in $X_{B_{4}}$. Clearly $e(G_{4})<e(G_{6})$. One can see that $G_{6}$ has no cycle of length more than $2c$. Thus, $C$ is a longest cycle of $G_{6}$ as well, and $\rho_{G_{4}}(G_{4}-C)\leq\rho_{G_{6}}(G_{6}-C)$, a contradiction to (ii). Hence we conclude $|Y_{B_{4}}|=c$.

This implies that $G_{4}$ is isomorphic to $L_{a,b}$ or $L_{b,a}$. In any case, we have $\rho_{G_{4}}(G_{4}-C)=\varrho(a,b)$, a contradiction. This proves Claim 1.

Next we distinguish the following cases and derive a contradiction for each one.

Case 1**.**

$2c<b\leq a$.

For this case, $\varrho(a,b)=\varrho(a-1,b)+c=\varrho(a,b-1)+c$. If there is a vertex in $G-C$ with degree at most $c$, then by the choice of $G$, $\rho_{G}(G-C)\leq\varrho(a,b)$. Thus, we assume that every vertex in $G-C$ has degree at least $c+1$. Let $H$ be a component of $G-C$. By Lemma 2.7, for each vertex $v\in H$, there is an $(v,C)$-fan $F$ with $e(F)\geq c+1$. This implies $|V(C)|\geq 2c+2$, a contradiction.

Case 2**.**

$c\leq b\leq 2c<a$.

For this case, $\varrho(a,b)=\varrho(a-1,b)+c$. If there is a vertex in $X_{G-C}$ with degree at most $c$, then we can get a contradiction by the choice of $G$. So assume that every vertex in $X_{G-C}$ has degree at least $c+1$. Let $X^{\prime}\subseteq X_{G-C}$ with $|X^{\prime}|=c+1$, and $G^{\prime}=G[X^{\prime}\cup Y]$. Observe that for every $x\in X^{\prime}$, $d_{G^{\prime}}(x)\geq c+1=\max\{|X^{\prime}|,|Y|/2+1\}$. By Lemma 2.9, $G^{\prime}$ has a cycle containing all vertices in $X^{\prime}$, i.e., $G$ has a cycle of length $2c+2$, a contradiction.

Case 3**.**

$c\leq b<a\leq 2c$.

For this case, $\varrho(a,b)=\varrho(a-1,b)+c$. So every vertex in $X_{G-C}$ has degree at least $c+1$. Since $a>b$, we can choose $H$ as a component of $G-C$ with $|X_{H}|>|Y_{H}|$. Let $N_{C}(H)=\{u_{1},u_{2},\cdots,u_{\alpha}\}$, where $u_{1},\cdots,u_{\alpha}$ appear in this order along $C$. Let $v_{1}\in N_{H}(u_{1})$. By Lemma 2.1, $H$ has a maximal $v_{1}$-path $P$ with terminus in $X_{H}$. Let $s\in X_{H}$ be the terminus of $P_{1}$. We extend the path $sP_{1}v_{1}u_{1}\overleftarrow{C}[u_{1},u_{2}]$ to be a maximal $t$-path, say $P$. Thus $P$ is a maximal path of $G$. Let $t$ be the end-vertex of $P$ other than $s$. Since $d(s)\geq c+1$, we have $|V(P)|\geq 2c+2$ and $d(s)+d(t)\geq c+3$. By Lemma 2.8, $G$ has a cycle of length more than $2c$, a contradiction.

Case 4**.**

$c\leq b=a\leq 2c$.

For this case, $\varrho(a,b)=\varrho(a-1,b-1)+c+1$. If there is a pair of vertices $(x,y)\in(X_{G-C},Y_{G-C})$ with $\rho(x,y)\leq c+1$, then we can prove by the choice of $G$. So assume that for every $(x,y)\in(X_{G-C},Y_{G-C})$, $\rho(x,y)\geq c+2$. Recall that

[TABLE]

Case 4.1**.**

$G-C$ is disconnected.

First assume that there is a balanced component $H$ of $G-C$. Then both $G_{1}=G[V(C)\cup V(H)]$ and $G_{2}=G-H$ are balanced. Clearly, $C$ is a longest cycle of $G_{1}$ and $G_{2}$ as well. Since $|G_{1}|<|G|$, we get

[TABLE]

and similarly,

[TABLE]

Thus

[TABLE]

Now we assume that every component of $G-C$ is not balanced. Let $H_{1},H_{2}$ be two components of $G-C$ such that $|X_{H_{1}}|>|Y_{H_{1}}|$, $|X_{H_{2}}|<|Y_{H_{2}}|$. Let $N_{C}(H_{1})\cup N_{C}(H_{1})=\{u_{1},u_{2},\ldots,u_{\alpha}\}$, where the vertices appear in this order along $C$. Set $\beta_{1}=\min\{d_{H_{1}}(x):x\in X_{H_{1}}\}$ and $\beta_{2}=\min\{d_{H_{2}}(y):y\in Y_{H_{2}}\}$. Thus $\alpha+\beta_{1}+\beta_{2}\geq c+2$.

Without loss of generality, we suppose $u_{1}\in N_{C}(H_{1})$ and $u_{2}\in N_{C}(H_{2})$. Let $v_{1}\in N_{H_{1}}(u_{1})$, $v_{2}\in N_{H_{2}}(u_{2})$. By Lemma 2.1, $H_{1}$ has a maximal $v_{1}$-path $P_{1}$ with terminus in $X_{H_{1}}$ and of order at least $2\beta_{1}$, $H_{2}$ has a maximal $v_{2}$-path $P_{2}$ with terminus in $Y_{H_{2}}$ and of order at least $2\beta_{2}$. Thus, $P=P_{1}v_{1}\overleftarrow{C}[u_{1},u_{2}]v_{2}P_{2}$ is a maximal path of $G$. Let $s,t$ be the ends of $P$. Recall that $d(s)+d(t)\geq c+2$. If $|V(P)|\geq 2c+1$, then by Lemma 2.8, $G$ has a cycle of length more than $2c$, a contradiction. So assume that $|V(P)|\leq 2c$. This implies that $\ell(\overrightarrow{C}[u_{1},u_{2}])\geq 2$. Therefore, we infer $\ell(\overrightarrow{C}[u_{i},u_{i+1}])\geq 2$ for all $i=1,2,\cdots,\alpha$ (the index are taken modular $\alpha$). Moreover,

[TABLE]

Thus we have $\ell(\overrightarrow{C}[u_{1},u_{2}])\geq 2\beta_{1}+2\beta_{2}+1$. So

[TABLE]

a contradiction.

Case 4.2**.**

$G-C$ is connected.

Let $H=G-C$, $N_{C}(H)=\{u_{1},u_{2},\ldots,u_{\alpha}\}$, and $\beta=\min\{\rho_{H}(x,y):x\in X_{H},y\in Y_{H}\}$. So $\alpha+\beta\geq c+2$. Note that $H$ is balanced.

Case 4.2.1**.**

$N_{C}(H)\cap X_{C}\neq\emptyset$ and $N_{C}(H)\cap Y_{C}\neq\emptyset$.

Without loss of generality, we assume that $u_{1}\in X_{C}$, $u_{2}\in Y_{C}$. Let $v_{1}\in N_{H}(u_{1})$, $v_{2}\in N_{H}(u_{2})$. By Lemma 2.2, $H$ has a detached maximal $(v_{1},v_{2})$-DPP $D$ with order at least $\beta+1$. Thus $P=D\cup v_{1}u_{1}\overleftarrow{C}u_{2}v_{2}$ is a maximal path of $G$. Let $s,t$ be the two end-vertices of $P$. So $d(s)+d(t)\geq c+2$. If $|V(P)|\geq 2c+1$, then by Lemma 2.8, $G$ has a cycle of length more than $2c$, a contradiction. So we derive $|V(P)|\leq 2c$. Therefore,

[TABLE]

This implies $\ell(\overrightarrow{C}[u_{1},u_{2}])\geq\beta+2$.

Note that there also exists some subscript $i$ such that $u_{i}\in Y_{C}$ and $u_{i+1}\in X_{C}$. We can get $\ell(\overrightarrow{C}[u_{i},u_{i+1}])\geq\beta+2$ as above. So

[TABLE]

a contradiction.

Case 4.2.2**.**

$N_{C}(H)\cap X_{C}=\emptyset$ or $N_{C}(H)\cap Y_{C}=\emptyset$.

Without loss of generality, we assume that $N_{C}(H)\subseteq Y_{C}$. Recall that $N_{C}(H)=\{u_{1},u_{2},\ldots,u_{\alpha}\}$, and $\beta=\min\{\rho_{H}(x,y):x\in X_{H},y\in Y_{H}\}$. Since $G$ is 2-connected, $\alpha\geq 2$ and $|N_{H}(C)|\geq 2$. Specially, $|X_{H}|=|Y_{H}|\geq 2$.

First assume that $H$ is 2-connected. Then at least two of $u_{i}$ have the property that $u_{i},u_{i+1}$ are adjacent to two distinct vertices in $H$. If $u_{i},u_{i+1}$ are adjacent to two distinct vertices in $H$, then by Lemma 2.6, there is a $(u_{i},u_{i+1})$-path of length at least $\beta+1$ with all internal vertices in $H$. It follows that $\ell(\overrightarrow{C}[u_{i},u_{i+1}])\geq\beta+1$. Recall that there are at least two pairs $\{u_{i},u_{i+1}\}$ that have two distinct neighbors in $H$. Thus

[TABLE]

a contradiction.

Now we assume that $H$ has connectivity 1. Then at least two of $u_{i}$ have the property that $u_{i},u_{i+1}$ are adjacent to a good pair of $H$. If $u_{i},u_{i+1}$ are adjacent to a good pair of $H$, say $\{x_{i},x_{i+1}\}$, then by Lemma 2.3, $H$ has a detached maximal $\{x_{i},x_{i+1}\}$-DPP $D$ of order at least $\beta+1$. Let $s,t$ be the termini of the two paths of $D$. It follows that $P=D\cup(x_{1}u_{1}\overleftarrow{C}u_{2}x_{2})$ is a maximal path of $G$. If $|V(P)|\geq 2c+1$, then $G$ has a cycle of length more than $2c$ by Lemma 2.8. So assume that $|V(P)|\leq 2c$. That is,

[TABLE]

This implies $\ell(\overrightarrow{C}[u_{i},u_{i+1}])\geq\beta+2$.

Recall that there are at least two pairs $\{u_{i},u_{i+1}\}$ that are adjacent to a good pair of $H$. It follows that

[TABLE]

a contradiction. The proof is complete.

3 Proofs of Theorems 1.4 and 1.5

In this section, we prove Theorem 1.5 and then prove Theorem 1.4. We first give a sketch of the proof of Theorem 1.5. We shall show that $G$ is weakly bipancyclic with girth 4 and prove it by contradiction. Suppose not. Then for a longest cycle $C$, the hamiltonian graph $G[C]$ is not bipancyclic. By a theorem of Entringer and Schmeichel [8], we can get an upper bound of $e(G[C])$. Notice that Theorem 1.7 gives an upper bound of $e(G-C)+e(G-C,C)$. Finally, this can give us an estimate of an upper bound of $e(G)$, which contradicts the edge number condition.

Proof of Theorem 1.5 needs the following.

Theorem A** (Entringer and Schmeichel [8]).**

Let G be a hamiltonian bipartite graph of order $2n\geq 8$. If $e(G)>\frac{n^{2}}{2}$, then $G$ is bipancyclic.

Now we are in stand for giving the proof of Theorem 1.5.

Proof of Theorem 1.5. (i) Set $n=m$ and $t=n-k$ in Theorem 1.1. Since $n\geq 2k+2$, we have $n\leq 2t-2$. By computation, we get $e(G)\geq n(n-k-1)+k+2>n+(n-1)(t-1)$. By Theorem 1.1, $c(G)\geq 2t=2(n-k)$. The proof is complete.

(ii) Suppose that $G$ is not weakly bipancyclic with girth 4. Let $C$ be a longest cycle in $G$, $2c$ the length of $C$ in $G$, and $G^{\prime}=G[C]$. Notice that $G^{\prime}$ is hamiltonian. Since $G$ is not weakly bipancyclic with girth 4, $G^{\prime}$ is not bipancyclic. By Theorem A, we have $e(G^{\prime})\leq\frac{c^{2}-1}{2}$. By Theorem 1.5(i), we know $c\geq n-k\geq k+2$ where $k\in\mathbb{Z}$, and $2c-n\geq n-2k>0$. By Theorem 1.7,

[TABLE]

Recall that $n-k\leq c\leq n$. Set a function $f(x)=-\frac{x^{2}}{2}+x(n-1)+n-\frac{1}{2}$, where $n-k\leq x\leq n$. The symmetric axis is $x=n-1$. First, suppose that $k=0$ or $k\geq 2$. Notice that $n-(n-1)\leq(n-1)-(n-k)$. There holds

[TABLE]

since $n\geq 2(k+1)$, a contradiction. Let $k=1$. Then $f(n-1)=\frac{(n-1)^{2}}{2}+n-\frac{1}{2}<n(n-2)+3$, a contradiction. The proof is complete.

Using Theorem 1.5, we can prove Theorem 1.4 now.

Proof of Theorem 1.4. We prove the theorem by contradiction. Let $G$ be a counterexample such that the order $n+m$ is smallest, and then the number of edges is smallest. By Theorem 1.5, we know $n\geq m+1$. If there is a vertex, say $y\in Y$, such that $d_{X}(y)\leq m-k-1$, then let $G^{\prime}:=G-y$. We can see $e(G^{\prime})=e(G)-d_{X}(y)\geq(n-1)(m-k-1)+k+2$. By the choice of $G$, $G^{\prime}$ is not a counterexample, and so is $G$, a contradiction. Thus, $d_{X}(y)\geq m-k$ for each vertex $y\in Y$. Hence $e(G)\geq n(m-k)>(n-1)(m-k-1)+k-2$, since $m\geq 2k+2$ and $n\geq 2$. Now we can delete an edge in $G$ and find a smaller counterexample, a contradiction. This contradiction completes the proof.

Acknowledgements

Part of writing was done when the second author was visiting Fengming Dong at NTU, Singapore. The second author is grateful to all his help, enthusiasm, and encouragement during his visit. The second author is also grateful to Jie Ma for sharing his knowledge on this topic.