A Best Possible Result for the Square of a 2-Block to be Hamiltonian

TL;DR

This paper proves the optimal condition under which the square of a 2-block graph contains a Hamiltonian cycle passing through four specified vertices and edges, extending understanding of Hamiltonian properties in graph squares.

Contribution

It establishes the best possible result for the existence of Hamiltonian cycles in the square of a 2-block graph with four specified vertices and edges.

Findings

Existence of Hamiltonian cycle in G^2 passing through four specified edges

Result is proven to be optimal

Applicable to 2-block graphs of order greater than 3

Abstract

It is shown that for any choice of four different vertices x_1,...,x_4 in a 2-block G of order p>3, there is a hamiltonian cycle in G^2 containing four different edges x_iy_i of E(G) for certain vertices y_i, i=1,2,3,4. This result is best possible.

Equations

$$H_{G_1}=(H_B-y^{\prime }y)\cup y^{\prime }{y}^{\ast }$$

$$H_{G_1}=(H_B\cup C-\{y^{\prime }y,y^{\prime \prime }y\})\cup y^{\prime }{y}^{\prime \prime }$$

$$(H_{G_1}\cup H_{B^{\ast }}-\{y_2^{\ast }{y}^{\ast },y_1^{\ast }{y}^{\ast }\})\cup \{y_1^{\ast }{y}_2^{\ast }\}$$

$$(H_{G_1}\cup H_{G_2}\cup H_{B^{\ast }}-\{y_2^{\ast }{y}^{\ast },y_1^{\ast }{y}^{\ast },x^{\prime }x,x^{\prime \prime }x\})\cup \{y_1^{\ast }{y}_2^{\ast },x^{\prime \prime }{x}^{\prime }\}$$

$$H^{\ast }=(H_B-\{y{y}^{\ast },y{y}^{\ast \ast }\})\cup y^{\ast }{y}^{\ast \ast }$$

$$H=G\cup \{y_1,y_2,...,y_t;t\ge 3\}\cup \{x_i{y}_j:1\le i\le 2,1\le j\le t\},$$

Full text

Keywords: square of graphs, hamiltonian cycles

2010 Mathematics Subject Classification: 05C38,05C48

1 Introduction

As for standard terminology, we refer to the book by Bondy and Murty, [2], and to the papers quoted in the references.

The square of a graph $G$, denoted $G^{2}$, is the graph obtained from $G$ by joining any two nonadjacent vertices which have a common neighbor, by an edge. Fairly recent development in hamiltonian graph theory has shown a resurgence of interest in hamiltonian cycles and paths in the square of 2-connected graphs (which we call 2-blocks for short). In particular, short proofs have been found for two results of the second author of the present paper, [10], [11]. And more recently, in [1] the authors develop algorithms which are linear in $|E(G)|$ and produce a hamiltonian cycle, a hamiltonian path joining arbitrary vertices $u$ and $v$ respectively, in $G^{2}$. Moreover, they develop an algorithm running in $O(|V(G)|^{2})$ time and producing cycles of arbitrary length from 3 to $|V(G)|$.

Also very recently it was shown in [3] and [9] that a 2-block has the $\mathcal{F}_{4}$ property; that is, given vertices $x_{1},x_{2},x_{3},x_{4}$ in the 2-block $G$, there is a hamiltonian path in $G^{2}$ joining $x_{1}$ and $x_{2}$ and traversing distinct edges $x_{3}y_{3}$ and $x_{4}y_{4}$ of $G$ (see Theorem 7). The proof of this result is very long and is based on techniques developed by Fleischner in [5], [6], [7] and by Fleischner and Hobbs in [8]. It remains to be shown whether one can find a much shorter proof of this result. However, this result will be of importance in the proof of the main result of the current paper.

We start with a definition.

Definition 1**.**

A graph $G$ is said to have the $\mathcal{H}_{k}$ property if for any given vertices $x_{1},...,x_{k}$ there is a hamiltonian cycle in $G^{2}$ containing distinct edges $x_{1}y_{1},...,x_{k}y_{k}$ of $G$.

We note in passing that $G$ having the $\mathcal{F}_{4}$ property implies that $G$ has the $\mathcal{H}_{3}$ property; clearly, choose $x_{1}$, $x_{2}$, $x_{3}$ arbitrarily and a different $x_{4}$ adjacent to some $x_{1}$ for $i\in\{1,2,3\}$ in $G$, say $i=1$. A hamiltonian path in $G^{2}$ joining $x_{1}$ and $x_{4}$ and containing edges $x_{2}y_{2}$ and $x_{3}y_{3}$ of $G$ yields a hamiltonian cycle containing these two edges of $G$ and $x_{1}x_{4}$ which lies also in $G$.

The main result of this paper is the following.

Theorem 2**.**

Given a 2-block $G$ on at least four vertices, then $G$ has the $\mathcal{H}_{4}$ property, and there are 2-blocks of arbitrary order greater than 4 without the $\mathcal{H}_{5}$ property.

This theorem and the $\mathcal{F}_{4}$ property of 2-blocks are key to describe the most general block-cut vertex structure a graph $G$ may have in order to guarantee that $G^{2}$ is hamiltonian, hamiltonian connected, respectively. This will be done in follow-up papers.

Moreover Theorem 2 gives the positive answer to Conjecture 5.4 stated in [4] as an immediate corollary.

Corollary 3**.**

Let $G$ be a connected graph such that its block-cutvertex graph $\mbox{bc}(G)$ is homeomorphic to a star in which the center $c$ corresponds to a block $B_{c}$ of $G$. If $B_{c}$ contains at most 4 cutvertices, then $G^{2}$ is hamiltonian.

2 Preliminaries

However, before proving Theorem 2 we mention several concepts and results which we need to make use of, and we prove a lemma.

A graph $G$ is an edge-critical block, if $\kappa(G)=2$ and $\kappa(G-e)=1$ for any edge $e$ of $G$ . Let $D(G)$ be the set of edges $uv$ where $d_{G}(u),d_{G}(v)\geq 3$. If $D(G)=\emptyset$, then every edge of $G$ is incident to a vertex of degree 2; we call such a graph a DT-graph.

Theorem 4**.**

[6**]***

Let $G$ be an edge-critical block. Then exactly one of the following two statements is true:*

$G$* is a DT-block.*

• 2)

There is an edge $f$ in $D(G)$ such that at least one of the endblocks of $G-f$ is a DT-block.

The basic result about hamiltonicity of the square of a 2-block is given by the following theorem.

Theorem 5**.**

[7**]***

Suppose $v$ and $w$ are two arbitrarily chosen vertices of a $2$-block $G$. Then $G^{2}$ contains a hamiltonian cycle $C$ such that the edges of $C$ incident to $v$ are in $G$ and at least one of the edges of $C$ incident to $w$ is in $G$. Furthermore, if $v$ and $w$ are adjacent in $G$, then these are three different edges.*

Let $\mbox{bc}(G)$ denote the block-cutvertex graph of $G$. Blocks corresponding to leaves of $\mbox{bc}(G)$ are called endblocks. Note that a block in a graph $G$ is either a 2-block or a bridge of $G$. The graph $G$ is called blockchain if $\mbox{bc}(G)$ is a path. Let $G$ be a blockchain. We denote its blocks $B_{1},B_{2},...,B_{k}$ and cutvertices $c_{1},c_{2},...,c_{k-1}$ such that $c_{i}\in V(B_{i})\cap V(B_{i+1})$, for $i=1,2,...,k-1$. A blockchain $G$ is called trivial, if $E(\mbox{bc}(G))=\emptyset$, otherwise it is called non-trivial. Note that only $B_{1}$ and $B_{k}$ are endblocks of a non-trivial blockchain $G$. An inner block is a block of $G$ containing exactly 2 cutvertices. An inner vertex is a vertex in $G$ which is not a cutvertex of $G$.

The first author proved in [4] the following theorem dealing hamiltonicity of the square of a blockchain graph.

Theorem 6**.**

[4**]***

Let $G$ be a blockchain and let $u_{1}$, $u_{2}$ be arbitrary inner vertices which are contained in different endblocks of $G$.

Then $G^{2}$ contains a hamiltonian cycle $C$ such that, for $i=1,2$,*

• $\bullet$

if $u_{i}$ is contained in a 2-block, then both edges of $C$ incident with $u_{i}$ are in $G$, and

• $\bullet$

if $u_{i}$ is not contained in a 2-block, then exactly one edge of $C$ incident with $u_{i}$ is in $G$.

Let $G$ be a connected graph. By a $uv$-path we mean a path from $u$ to $v$ in $G$. If a $uv$-path is hamiltonian, we call it a $uv$-hamiltonian path. Let $A=\{x_{1},x_{2},...,x_{k}\}$ be a set of $k$ ($\geq 3$) distinct vertices in $G$. An $x_{1}x_{2}$-hamiltonian path in $G^{2}$ which contains $k-2$ distinct edges $x_{i}y_{i}\in E(G),i=3,...,k$, is said to be $\mathcal{F}_{k}$. A graph $G$ is said to have the $\mathcal{F}_{k}$ property if, for any set $A=\{x_{1},x_{2},...,x_{k}\}\subseteq V(G)$, there is an $\mathcal{F}_{k}$ $x_{1}x_{2}$-hamiltonian path in $G^{2}$.

Theorem 7**.**

[9**]***

Let $G$ be a 2-block. Then $G$ has the $\mathcal{F}_{4}$ property.*

A graph $G$ is said to have the strong $\mathcal{F}_{3}$ property if, for any set of 3 vertices $\{x_{1},x_{2},x_{3}\}$ in $G$, there is an $x_{1}x_{2}$-hamiltonian path in $G^{2}$ containing distinct edges $x_{3}z_{3},x_{i}z_{i}\in E(G)$ for a given $i\in\{1,2\}$. Such an $x_{1}x_{2}$-hamiltonian path in $G^{2}$ is called a strong $\mathcal{F}_{3}$ $x_{1}x_{2}$-hamiltonian path.

Theorem 8**.**

[9**]***

Every 2-block has the strong $\mathcal{F}_{3}$ property.*

The following lemma is frequently used in the proofs below.

Lemma 9**.**

Let $G$ be a non-trivial blockchain. We choose

• •

$c_{0}\in V(B_{1})$, $c_{k}\in V(B_{k})$ which are not cutvertices;

• •

$u_{i}\in V(B_{i})$* (if any) which is not a cutvertex and $v_{i}\in V(B_{i})$ such that $u_{i}\neq v_{i}$, $u_{1}\neq c_{0}$ and $u_{k}\neq c_{k}$, for $i=1,2,...k$.*

Then $G^{2}$ contains a $c_{0}c_{k}$-hamiltonian path $P$ such that there exist distinct edges $u_{i}u^{\prime}_{i}$ $v_{i}v^{\prime}_{i}\in E(B_{i})\cap E(P)$ (if $u_{i}$ exists), $i=1,2,...,k$.

Proof.

If $B_{i}$ is 2-connected, then let $P_{i}$ be an $\mathcal{F}_{4}$ $c_{i-1}c_{i}$-hamiltonian path in $B_{i}^{2}$ containing 2 distinct edges $u_{i}u^{\prime}_{i},v_{i}v^{\prime}_{i}\in E(B_{i})$ for $v_{i}\notin\{c_{i-1},c_{i}\}$ by Theorem 7; and let $P_{i}$ be a strong $\mathcal{F}_{3}$ $c_{i-1}c_{i}$-hamiltonian path in $B_{i}^{2}$ containing 2 distinct edges $u_{i}u^{\prime}_{i},v_{i}v^{\prime}_{i}\in E(B_{i})$ for $v_{i}\in\{c_{i-1},c_{i}\}$ by Theorem 8, respectively.

If $B_{i}=c_{i-1}c_{i}$, then we set $P_{i}=B_{i}$. Note that in this case $u_{i}$ does not exist and $v_{i}\in\{c_{i-1},c_{i}\}$.

Then $P=\cup_{i=1}^{k}P_{i}$ is a $c_{0}c_{k}$-hamiltonian path in $G^{2}$ as required. ∎

The concept of EPS-graphs plays a central role in proofs of hamiltonicity in the square of a $DT$-graph (see [5]). We use this concept also in one part of the proof of Theorem 2. Let $G$ be a graph. An EPS-graph is a spanning connected subgraph $S$ of $G$ which is the edge-disjoint union of an eulerian graph $E$ (which may be disconnected) and a linear forest $P$. For $S=E\cup P$, let $d_{E}(v)$, $d_{P}(v)$ denote the degree of $v$ in $E$, $P$, respectively.

Fleischner and Hobbs introduced in [8] the concept of $W$-soundness of a cycle. Let $W$ be a set of vertices of $G$. A cycle $K$ is called $W$-maximal if $|V(K^{\prime})\cap W|\leq|V(K)\cap W|$ for any cycle $K^{\prime}$ of $G$. Let $K$ be a cycle of $G$ and let $W$ be a set of vertices of $G$. A blockchain $P$ of $G-K$ is a $W$-separated $K$-to-$K$ blockchain based on vertex $x$ if a vertex of $W$ is a cut vertex of $P$, both endblocks $B$ and $B^{\prime}$ of $P$ include vertices of $K$, $V(B)\cap V(K)=\{x\}$, no vertex of $K$ is a cutvertex of $P$, and $(V(P)\cap V(K))-\{x\}\subseteq V(B^{\prime}).$ For a given path $p=v_{1},v_{2},...,v_{n-1},v_{n}$ we let $F(p)=v_{1}$, $L(p)=v_{n}$.

Definition 10**.**

A cycle $K$ in $G$ is $W$-sound if it is $W$-maximal, $|W|=5$ and the following hold:

• (1)

$|V(K)\cap W|\geq 4$; or

• (2)

$|V(K)\cap W|=3$* and the following situation does not prevail; there are two $W$-separated $K$-to-$K$ blockchains $P$ and $Q$ of $G-K$ based on a vertex $w$ of $W$ such that $V(P)\cap V(Q)=\{w\}$ and if $p$ is a shortest path in $P$ from $w$ to a vertex of $K$ different from $w$ and $q$ is the same for $Q$, then there is a subsequence $w,w^{\prime},L(p),L(q),w^{\prime\prime},w$ of $K$ where $w^{\prime}$ and $w^{\prime\prime}$ are in $W-\{w\}$; or*

• (3)

$|V(K)\cap W|=2$* and the following situation does not prevail; there are three $W$-separated $K$-to-$K$ blockchains $P_{1},P_{2}$ and $P_{3}$ of $G-K$ based on a single vertex $a$ of $V(K)-W$, such that $V(P_{i})\cap V(P_{j})=\{a\}$ whenever $i$ and $j$ are distinct elements of $\{1,2,3\}$, and if $p_{i}$ is a shortest path in $P_{i}$ from $a$ to a vertex of $K$ different from $a$ for each $i\in\{1,2,3\}$, then there is a subsequence $a,w^{\prime},L(p_{1}),L(p_{2}),L(p_{3}),w^{\prime\prime},a$ of $K$ where $\{w^{\prime},w^{\prime\prime}\}=V(K)\cap W$.*

We observe that Definition 10 is basically the content of Lemma 1 in [8]. That is, said lemma guarantees that for every choice $W\subseteq V(G)$ with $|W|=5$ in a 2-block $G$ of order at least 5, there is a $W$-sound cycle in $G$.

Theorem 11**.**

[8**]***

Let $G$ be a 2-block and $W$ a set of five distinct vertices in $G$, and let $K$ be a $W$-sound cycle in $G$. Then there is an EPS-graph $S=E\cup P$ of $G$ such that $K\subseteq E$ and $d_{P}(w)\leq 1$ for every $w\in W$.*

3 Proof of Theorem 2

Proof.

First we prove that $G$ has the $\mathcal{H}_{4}$ property. We proceed by contradiction supposing that $|V(G)|+|E(G)|$ is minimal. It follows that $G$ is an edge-critical block and in particular $|V(G)|\geq 5$. We distinguish cases by the number of edges in $D(G)$. The reader is advised to draw figures where he/she deems it necessary to follow our case distinctions.

Case 1. $|D(G)|>0.$ By Theorem 4, let $f=x^{\prime}x\in D(G)$ be an edge such that $d_{G}(x^{\prime}),d_{G}(x)\geq 3$. Then $G-f$ is a blockchain and both endblocks $B^{\prime},B$ of $G-f$ are 2-blocks. Set $X=\{x_{1},x_{2},x_{3},x_{4}\}$. Without loss of generality assume that $|X\cap(V(B)-y)|\leq 2$ (otherwise we consider $B^{\prime}$ instead of $B$); i.e., at most $x_{1},x_{2}\in V(B)-y$, say, where $x,y\in V(B)$ and $y$ is a cutvertex of $G-f$. We distinguish the following 3 subcases.

Subcase 1.1: $|X\cap(V(B)-y)|=2$; i.e., $x_{1},x_{2}\in V(B)-y.$

Then $B^{2}$ has an $xy$-hamiltonian path $P_{1}$ containing different edges $x_{1}y_{1}$, $x_{2}y_{2}$ of $E(G)$ for certain $y_{1},y_{2}$ by Theorem 7 or by Theorem 8 if $x_{1}=x$ or $x_{2}=x$; and $(G-B)^{2}$ has an $xy$-hamiltonian path $P_{2}$ containing different edges $x_{3}y_{3},x_{4}y_{4}$ of $E(G)$ for certain $y_{3},y_{4}$ by Lemma 9. Now $P_{1}\cup P_{2}$ is a required hamiltonian cycle in $G^{2}$, a contradiction. Note that $x_{3},x_{4}\in V(B^{\prime})-y^{\prime}$ where $y^{\prime}\in V(B^{\prime})$ is a cutvertex of $G-f$, otherwise we can use $B^{\prime}$ instead of $B$ and $x_{3}$ or $x_{4}$ instead of $x_{1}$ or $x_{2}$ (see Subcase 1.2 or Subcase 1.3 below).

Subcase 1.2: $|X\cap(V(B)-y)|=1$; i.e., $x_{1}\in V(B)-y$ and $x_{2}\notin V(B)-y.$

(1.2.1) Assume that $x_{2},x_{3},x_{4}$ are not inner vertices of $G$ in the same block of $G-B$. We proceed very similar as in Subcase 1.1; we use only the strong $\mathcal{F}_{3}$ property in $B$, and $G-B$ is a non-trivial blockchain. Hence we can apply Lemma 9 except if $x=x_{1}$, some $x_{i}=y$ for $i\in\{2,3,4\}$, say $i=2$, and $x_{3},x_{4}$ are inner vertices in the same endblock of $G-B$ which also contains $x_{2}$.

If $x=x_{1}$, $x_{2}=y$, and $x_{3},x_{4}$ are inner vertices in the same endblock of $G-B$ which also contains $x_{2}$, then $B^{2}$ has an $x_{2}x_{1}$-hamiltonian path $P_{1}$ containing different edges $x_{2}y_{2}$, $uv$ of $E(G)$ for certain $y_{2},u,v$ by Theorem 8, and $(G-B)^{2}$ has an $x_{2}x_{1}$-hamiltonian path $P_{2}$ containing different edges $x_{1}x^{\prime},x_{3}y_{3},x_{4}y_{4}$ of $E(G)$ for certain $y_{3},y_{4}$ by Lemma 9. Again, $P_{1}\cup P_{2}$ is a required hamiltonian cycle in $G^{2}$, a contradiction.

(1.2.2) Assume that $x_{2},x_{3},x_{4}$ are inner vertices of $G$ in the same block $B^{*}$ of $G-B$.

Clearly, $B^{2}$ contains a hamiltonian cycle $H_{B}$ containing 3 different edges $y^{\prime}y,x^{\prime}_{1}x_{1},x^{\prime\prime}x$ of $E(B)$ for certain vertices $y^{\prime},x^{\prime}_{1},x^{\prime\prime}$ by Theorem 7 (starting with a corresponding $\mathcal{F}_{4}$ $x^{\prime\prime}x$-hamiltonian path in $B^{2}$) if $x\neq x_{1}$, and $y^{\prime}y,x^{\prime}_{1}x,x^{\prime\prime}x$ of $E(B)$ for certain vertices $y^{\prime},x^{\prime}_{1},x^{\prime\prime}$ by Theorem 5 if $x=x_{1}$.

Let $G_{1}$ be the component of $G-B^{*}-xx^{\prime}$ containing $B$ and $y^{*}=V(B^{*})\cap V(G_{1})$. Note that $G_{1}$ is a trivial or non-trivial blockchain.

(a) If $y^{*}=y$, then $G_{1}=B$ and we set $H_{G_{1}}=H_{B}$ (see above).

(b) If $y^{*}\neq y$, then either $G_{1}-B=y^{*}y$ or $(G_{1}-B)^{2}$ contains a hamitonian cycle $C$ containing edges $y^{*}_{1}y^{*}$, $y^{\prime\prime}y$ of $E(G_{1}-B)$ for certain $y^{*}_{1},y^{\prime\prime}$ by applying Theorem 5 or Theorem 6.

Now we set

[TABLE]

and $y_{1}^{*}=y$ if $G_{1}-B=y^{*}y$; and

[TABLE]

if $G_{1}-B\neq y^{*}y$.

Note that the edge $y_{1}^{*}y^{*}\in E(G_{1})$ is contained in $H_{G_{1}}$ in both cases.

Clearly, $|V(B^{*})|+|E(B^{*})|<|V(G)|+|E(G)|$. Hence $(B^{*})^{2}$ contains a hamiltonian cycle $H_{B^{*}}$ containing four different edges $y_{2}^{*}y^{*},x_{2}x^{\prime}_{2},x_{3}x^{\prime}_{3},x_{4}x^{\prime}_{4}$ of $E(B^{*})$ for certain vertices $y_{2}^{*},x^{\prime}_{i}$, $i=2,3,4$.

Let $z\in V(B^{*})$ be the cutvertex of $G-x^{\prime}x$ different from $y^{*}$.

(A) $x^{\prime}=z$. Then

[TABLE]

is a required hamiltonian cycle in $G^{2}$ containing four different edges $x_{i}x^{\prime}_{i},$ of $E(G)$, $i=1,2,3,4$, a contradiction.

(B) $x^{\prime}\neq z$

If $d_{G-B^{*}}(z)=1$, then we set $G_{2}=G-G_{1}-B^{*}-z_{1}z$ where $z_{1}$ is the unique neighbour of $z$ in $G-B^{*}$; otherwise we set $G_{2}=G-G_{1}-B^{*}$. Note that $G_{2}$ is a trivial or non-trivial blockchain and $G_{2}=x^{\prime}x$ is not possible because of $d_{G}(x^{\prime})>2$.

We apply Theorem 6 such that either $(G_{2})^{2}$ contains a hamitonian cycle $H_{G_{2}}$ with $x^{\prime}x\in E(H_{G_{2}})$ if $z\notin V(G_{2})$, or $(G_{2})^{2}$ contains a hamitonian cycle $H$ containing the edge $x^{\prime}x$ and different edges $z_{1}z,z_{2}z$ of $G_{1}$ for certain $z_{1},z_{2}$ if $z\in V(G_{2})$. In the latter case we set $H_{G_{2}}=(H-\{z_{1}z,z_{2}z\})\cup z_{1}z_{2}$. Then

[TABLE]

is again a hamiltonian cycle in $G^{2}$ containing four different edges $x_{i}x^{\prime}_{i}$ of $E(G)$, $i=1,2,3,4$, a contradiction.

Subcase 1.3: $|X\cap(V(B)-y)|=0$; i.e., $x_{1},x_{2}\notin V(B)-y.$

Let $G_{1}$ be a graph which arises from $G$ by replacing $B$ with a path $p$ of length 3, say $p=x,a,b,y$. Then $|V(G_{1})|+|E(G_{1})|<|V(G)|+|E(G)|$ since $B$ is not a triangle because $G$ is edge-critical. Hence $(G_{1})^{2}$ contains a hamiltonian cycle $H_{1}$ containing four different edges $x_{i}y_{i}$ of $E(G_{1})$ for certain vertices $y_{i}$, $i=1,2,3,4$, and as many edges as possible of $G_{1}$.

In the following we shall proceed in a manner very similar to the proof that the square of a 2-block is hamiltonian, [6]. However, in order to avoid total dependence of the reader on the knowledge or study of [6], we shall describe and partially repeat the procedure employed in that paper. In particular, we shall quote the cases with the numbering of [6].

This yields the consideration of 13 cases how the hamiltonian cycle $H_{1}$ traverses vertices of the path $p$. As in [6], Cases 3, Case 4, Case 12, and Case 13 are contradictory to the maximality of the number of edges of $G_{1}$ belonging to $H_{1}$; and Case 6 can be reduced to Case 10, Case 8 to Case 7, Case 10 to Case 9 and Case 11 to Case 5. Note that by the reductions we preserve the existence of the edges $x_{i}y_{i}$ even if $x_{i}\in\{x^{\prime},y\}$ for $i\in\{1,2,3,4\}$.

The remaining 5 cases are (using the labeling of vertices $x^{\prime},x,a,b,y$ instead of $x,w,a,b,v$ in [6]):

Case 1. $H_{1}=...,x,a,b,y,...$

Case 2. $H_{1}=...,x,a,b,y^{\prime},...$

Case 5. $H_{1}=...,x^{\prime},a,b,x,...$

Case 7. $H_{1}=...,x^{\prime},a,y,...,y^{\prime},b,x$

Case 9. $H_{1}=...,x^{\prime},a,y,b,x...$;

and $y^{\prime}y$ is an edge of $G$.

In order to extend $H_{1}$ to $H$ in $G^{2}$ in these five cases with $H$ having the required property, one can proceed in the same way as it has been done in [6]. However, we deem it necessary to show explicitly that no problems arise under the stronger condition of this theorem (similarly as in [7]).

Case 1. By Theorem 8, $B^{2}$ has an $xy$-hamiltonian path $P$ starting with an edge $yy^{*}$ of $E(B)$ and containing an edge $uv$ of $B$ for certain vertices $u,v$. Replace in $H_{1}$ the path $p$ with a hamiltonian path $P$ and we get a hamiltonian cycle $H$ as required.

Case 2. Take $P$ as in Case 1 and replace in $H_{1}$ the path $x,a,b,y^{\prime}$ with $(P-yy^{*})\cup y^{\prime}y^{*}$ and again we get a hamiltonian cycle $H$ as required. Note that $H$ contains all edges of $G$ belonging to $H_{1}$.

Case 5. By Theorem 5, $B^{2}$ contains a hamiltonian cycle $H_{B}$ such that both edges of $H_{B}$ incident to $y$ (say $yy^{*},yy^{**}$) are in $B$ and at least one of the edges of $H_{B}$ incident to $x$ (say $xx^{*}$) is in $B$. We set

[TABLE]

which does not contain $y$, and replace in $H_{1}$ the path $x^{\prime},a,b,x$ with $(H^{*}-xx^{*})\cup x^{\prime}x^{*}$, thus obtaining a hamiltonian cycle $H$ in $G^{2}$ which has the same behavior in all vertices of $G_{1}-\{a,b\}\subset G$ as $H_{1}$.

Case 7. Take $H_{B}$ as in Case 5 and replace in $H_{1}$ the path $x^{\prime},a,y$ with the path $P_{1}\cup x^{*}x^{\prime}$ where $P_{1}\subset H_{B}$ is the path from $y$ to $x^{*}$ and does not contain $x$; and replace in $H_{1}$ the path $y^{\prime},b,x$ with the path $P_{2}\cup y^{\prime}t$ where $t\in\{y^{*},y^{**}\}$ and $P_{2}\subset H_{B}$ is the path from $x$ to $t$ and does not contain any of $y$, $x^{*}$. Again we get a hamiltonian cycle $H$ as required.

Case 9. Take $H_{B}$ as in Case 5 and replace in $H_{1}$ the path $x^{\prime},a,y,b,x$ with $(H_{B}-xx^{*})\cup x^{\prime}x^{*}$, thus obtaining a hamiltonian cycle $H$ in $G^{2}$ which has the same behavior in all vertices of $G_{1}-\{a,b,y\}\subset G$ as $H_{1}$ and both edges of $H$ incident to $y$ are in $G$.

In all cases we obtained a hamiltonian cycle $H$ in $G^{2}$ containing four different edges $x_{i}x^{\prime}_{i},$ of $E(G)$ (in most cases we have $x^{\prime}_{i}=y_{i}$; see the first paragraph of this subcase 1.3), $i=1,2,3,4$, a contradiction.

Case 2. $|D(G)|=0.$ That is, $G$ is a $DT$-graph.

a) Suppose $N(x_{i})\subseteq V_{2}(G)$ for every $i=1,2,3,4$.

Set $W^{\prime}=\{x_{1},x_{2},x_{3},x_{4}\}$ and let $K$ be a $W^{\prime}$-maximal cycle in $G$. Observe that $|V(K)|\geq 4$ since an edge-critical block on at least 4 vertices cannot contain a triangle.

If $|W^{\prime}\cap V(K)|=4$, then we choose $x_{5}$ arbitrary in $V(G)-W^{\prime}$. If $|W^{\prime}\cap V(K)|=3$, then we choose $x_{5}$ arbitrary in $V(K)-W^{\prime}$. If $|W^{\prime}\cap V(K)|=2$, then we choose an arbitrary 2-valent vertex $x_{5}$ in $V(K)-W^{\prime}$ which exists because all neighbours of $x_{i}$ are 2-valent.

We set $W=W^{\prime}\cup\{x_{5}\}$. Then $K$ is $W$-sound in $G$ unless $|W\cap V(K)|=3$ and forbidden situation (2) in Definition 10 arises. That is, without loss of generality $x_{1},x_{2}\in V(K)$ and there exist $W$-separated $K$-to-$K$ blockchains $P$, $Q$ based on $x_{i}$, $i\in\{1,2\}$, $P\cap Q=x_{i}$, and paths $p,q$ in $P,Q$, respectively, such that there is a subsequence $x_{i},w^{\prime},L(p),L(q),w^{\prime\prime},x_{i}$, where $\{w^{\prime},w^{\prime\prime}\}=\{x_{3-i},x_{5}\}$ and $x_{3},x_{4}\in V(p)\cup V(q)$. Then there is a cycle $K^{\prime}$ containing $x_{i},x_{3},x_{4}$, a contradiction to the $W^{\prime}$-maximality of $K$.

By Theorem 11, $G$ contains an EPS-graph $S=E\cup P$ such that $K\subseteq E$ and $d_{P}(w)\leq 1$ for every $w\in W$. If there is no adjacent pair $x_{i},x_{j}$ for $i,j\in\{1,2,3,4\}$, we use $S$ and an algorithm in [5] to obtain a hamiltonian cycle in $G^{2}$ with the required properties, a contradiction. However, if there is an adjacent pair, say $x_{1},x_{2}$, then $d_{G}(x_{1})=d_{G}(x_{2})=2$ and $d_{P}(x_{1})=d_{P}(x_{2})=0$ and we can proceed with the cycle $K$ containing $x_{1},x_{2},x_{3}$ to obtain a required hamiltonian cycle in $G^{2}$ as before, a contradiction.

b) Without loss of generality suppose that $N(x_{4})\nsubseteq V_{2}(G)$.

Hence $\mbox{deg}_{G}(x_{4})=2$. Let $P_{4}=y_{4}x_{4}z_{1}...z_{k}$ be a unique path in $G$ such that $d_{G}(y_{4})>2$, $d_{G}(z_{k})>2$ and $d_{G}(z_{i})=2$, for $i=1,2,...,k-1$. We set $G^{-}=G-\{x_{4},z_{1},...,z_{k-1}\}$, where $z_{0}=x_{4}$ if $k=1$.

b1) Assume that $G^{-}$ is 2-connected.

If $x_{i}\in V(G^{-})-\{y_{4},z_{k}\}$ for $i=1,2,3$, then $|V(G)|+|E(G)|>|V(G^{-})|+|E(G^{-})|$ and hence $(G^{-})^{2}$ has a hamiltonian cycle $H^{-}$ containing different edges $x_{i}y_{i},z_{k}w_{4}\in E(G)$, $i=1,2,3$. It is easy to see that we can extend $H^{-}$ to a hamiltonian cycle $H$ in $G^{2}$ such that $H$ contains edges $x_{i}y_{i}$, $x_{4}z_{1}$, for $i=1,2,3$, a contradiction.

Suppose $x_{3}\notin V(G^{-})-\{y_{4},z_{k}\}$. If $\{x_{1},x_{2},x_{3}\}\cap\{y_{4},z_{k}\}\neq\emptyset\}$, then without loss of generality $x_{3}\in\{y_{4},z_{k}\}$. By Theorem 7 or Theorem 8, $(G^{-})^{2}$ contains a $y_{4}z_{k}$-hamiltonian path $P^{-}$ and $P^{-}$ contains distinct edges $x_{i}y_{i}$ of $G$ if $x_{i}\in V(G^{-})$ for $i=1,2$. Then $P^{-}\cup P_{4}$ is a hamiltonian cycle in $G^{2}$ with the required properties, a contradiction.

b2) Assume that $G^{-}$ is not 2-connected.

Then $G^{-}$ is a non-trivial blockchain with $y_{4},z_{k}$ in distinct endblocks and $y_{4},z_{k}$ are not cutvertices.

Assume not all $x_{1},x_{2},x_{3}$ are inner vertices in the same block. Then we apply Lemma 9 to get a $y_{4}z_{k}$-hamiltonian path $P^{-}$ in $(G^{-})^{2}$ with distinct edges $x_{i}y_{i}\in E(G^{-})$, $i=1,2,3$. Note than $x_{i}$ could be $y_{4}$ or $z_{k}$. Then again $P^{-}\cup P_{4}$ is a hamiltonian cycle in $G^{2}$ with the required properties, a contradiction.

Now assume that $x_{1},x_{2},x_{3}$ are inner vertices in the same block $B$. Then there exists an end block $B^{*}$ of $G^{-}$ such that $x_{i}\notin V(B^{*})$, $i=1,2,3$. A graph $G^{\prime}$ arises from $G$ by the replacement of $B^{*}$ by a path $p$ of length 3. Hence $|V(G)|+|E(G)|>|V(G^{\prime})|+|E(G^{\prime})|$ and we denote by $H^{\prime}$ a hamiltonian cycle in $(G^{\prime})^{2}$ containing edges $x_{i}w_{i}$, $i=1,2,3,4$, and as many edges of $G^{\prime}$ as possible.

We proceed in the same manner as in Subcase 1.3 (note that in this case none of $x_{i}$, $i=1,2,3,4$, is on $p$) to get a hamiltonian cycle in $G^{2}$ with required properties, a contradiction.

Finally we want to show that Theorem 2 is best possible, i.e., we construct an infinite family of graphs which do not satisfy the $\mathcal{H}_{5}$ property. For this purpose start with an arbitrary 2-block $G$ and fix different vertices $x_{1},x_{2}\in V(G)$.

Define

[TABLE]

where $\{y_{1},...,y_{t}\}\cap V(G)=\emptyset$.

Then $H$ is a 2-block. However, $H$ does not have the $\mathcal{H}_{5}$ property: indeed, there is no hamiltonian cycle $C$ in $H^{2}$ containing edges of $H$ incident to $x_{1},x_{2},y_{1},y_{2},y_{3}$ because of the neighbours of $y_{1},y_{2},y_{3}$ in $H$ are $x_{1}$ and $x_{2}$ only; that is $x_{1}$ or $x_{2}$ would be incident to three edges of $C\cap H$, which is impossible. ∎

4 Conclusion

We introduced the concept of the $\mathcal{H}_{k}$ property and proved that every 2-block has the $\mathcal{H}_{4}$ property but not the $\mathcal{H}_{5}$ property in general. Similarly in [9] it is proved that every 2-block has the $\mathcal{F}_{4}$ property but not the $\mathcal{F}_{5}$ property in general. Moreover, a 2-block $G$ having the $\mathcal{F}_{k}$ property implies that $G$ has the $\mathcal{H}_{k-1}$ property for $k=3,4,...$. Hence we conclude that Theorem 2 and Theorem 7 are best possible with respect to hamiltonicity and hamiltonian connectedness in the square of a 2-block.

Acknowledgements.

This publication was supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports, and by FWF project P27615-N25.