Variable degeneracy on toroidal graphs

TL;DR

This paper investigates structural properties of planar and toroidal graphs to establish conditions for the existence of strictly $f$-degenerate transversals, leading to improved bounds on DP-colorability and vertex arboricity.

Contribution

It introduces new structural results and sufficient conditions for strictly $f$-degenerate transversals, enhancing understanding of DP-coloring and vertex arboricity in specific graph classes.

Findings

Toroidal graphs without certain subgraphs are DP-4-colorable.

Toroidal graphs without 4-cycles are DP-4-colorable.

Planar graphs without specific configurations are DP-4-colorable.

Abstract

DP-coloring was introduced by Dvo\v{r}\'{a}k and Postle as a generalization of list coloring and signed coloring. A new coloring, strictly $f$-degenerate transversal, is a further generalization of DP-coloring and $L$-forested-coloring. In this paper, we present some structural results on planar and toroidal graphs with forbidden configurations, and establish some sufficient conditions for the existence of strictly $f$-degenerate transversal based on these structural results. Consequently, (i) every toroidal graph without subgraphs isomorphic to the configurations in Fig.2 is DP-$4$-colorable, and has list vertex arboricity at most $2$, (ii) every toroidal graph without $4$-cycles is DP-$4$-colorable, and has list vertex arboricity at most $2$, (iii) every planar graph without subgraphs isomorphic to the configurations in Fig.3 is DP-$4$-colorable, and has list vertex arboricity at most $2$. These results improve upon previous results on DP-$4$-coloring [Discrete Math. 341~(7) (2018) 1983--1986; Bull. Malays. Math. Sci. Soc. 43~(3) (2020) 2271--2285] and (list) vertex arboricity [Discrete Math. 333 (2014) 101--105; Int. J. Math. Stat. 16~(1) (2015) 97--105; Iranian Math. Soc. 42~(5) (2016) 1293--1303].

Equations

$$f(v,1)+f(v,2)+\cdots +f(v,\kappa )={\mathrm{deg}}_B(v)$$

$$\sum_{v\,\in\,V(G)}\big{(}\deg(v)-4\big{)}+\sum_{f\,\in\,F(G)}\big{(}\deg(f)-4\big{)}=-4\epsilon(G)\leq 0.$$

$${\mu }^{\prime }(v)\ge \mathrm{deg}(v)-4-\lfloor \frac{\mathrm{deg}(v)}{3}\rfloor \times \frac{2}{3}>0.$$

$$V(G)=\mathcal{D}=\big{\{}\,v\mid f(v,1)+f(v,2)+\dots+f(v,\kappa)\geq\deg_{G}(v)=4\,\big{\}}.$$

$$\sum_{v\,\in\,V(G)}\big{(}\deg(v)-6\big{)}+\sum_{f\,\in\,F(G)}\big{(}2\deg(f)-6\big{)}=-6\epsilon(G)\leq 0.$$

$$V(F)\subseteq\mathcal{D}=\big{\{}\,v\mid f(v,1)+f(v,2)+\dots+f(v,\kappa)\geq\deg_{G}(v)=4\,\big{\}}.$$

$$\sum_{v\,\in\,V(G)}\big{(}\deg(v)-4\big{)}+\sum_{f\,\in\,F(G)}\big{(}\deg(f)-4\big{)}=-8.$$

$$V(F_{5})\subseteq\mathcal{D}=\big{\{}\,v\mid f(v,1)+f(v,2)+\dots+f(v,\kappa)\geq\deg_{G}(v)=4\,\big{\}}.$$

Full text

1 Introduction

All graphs considered in this paper are finite, undirected and simple. Let $\mathbb{Z}^{*}$ stand for the set of nonnegative integers, and let $f$ be a function from $V(G)$ to $\mathbb{Z}^{*}$. A graph $G$ is strictly $f$-degenerate if each nonempty subgraph $\Gamma$ of $G$ has a vertex $v$ such that $\deg_{\Gamma}(v)<f(v)$. A cover of a graph $G$ is a graph $H$ with vertex set $V(H)=\bigcup_{v\in V(G)}L_{v}$, where $L_{v}=\{\,(v,1),(v,2),\dots,(v,\kappa)\,\}$; the edge set $\mathscr{M}=\bigcup_{uv\in E(G)}\mathscr{M}_{uv}$, where $\mathscr{M}_{uv}$ is a matching between $L_{u}$ and $L_{v}$. Note that $L_{v}$ is an independent set in $H$ and $\mathscr{M}_{uv}$ may be an empty set. For convenience, this definition is slightly different from Bernshteyn and Kostochka’s [4], but consistent with Schweser’s [23]. A vertex subset $R\subseteq V(H)$ is a transversal of $H$ if $|R\cap L_{v}|=1$ for each $v\in V(G)$.

Let $H$ be a cover of $G$ and $f$ be a function from $V(H)$ to $\mathbb{Z}^{*}$, we call the pair $(H,f)$ a valued cover of $G$. Let $S$ be a subset of $V(G)$, we use $H_{S}$ to denote the induced subgraph $H[\bigcup_{v\in S}L_{v}]$. A transversal $R$ is a strictly $f$-degenerate transversal if $H[R]$ is strictly $f$-degenerate.

Let $H$ be a cover of $G$ and $f$ be a function from $V(H)$ to $\{0,1\}$. An independent transversal, or DP-coloring, of $H$ is a strictly $f$-degenerate transversal of $H$. It is observed that a DP-coloring is a special independent set in $H$. DP-coloring, also known as correspondence coloring, is introduced by Dvořák and Postle [13]. It is shown [21] that strictly $f$-degenerate transversal generalizes list coloring, $(f_{1},f_{2},\dots,f_{\kappa})$-partition, signed coloring, DP-coloring and $L$-forested-coloring.

The DP-chromatic number $\chi_{\mathrm{DP}}(G)$ of $G$ is the least integer $t$ such that $(H,f)$ has a DP-coloring whenever $H$ is a cover of $G$ and $f$ is a function from $V(H)$ to $\{0,1\}$ with $f(v,1)+f(v,2)+\dots+f(v,\kappa)\geq t$ for each $v\in V(G)$. A graph $G$ is DP-$m$-colorable if its DP-chromatic number is at most $m$.

Dvořák and Postle [13] presented a non-trivial application of DP-coloring to solve a longstanding conjecture by Borodin [8], showing that every planar graph without cycles of lengths $4$ to $8$ is $3$-choosable. Another application of DP-coloring can be found in [4], Bernshteyn and Kostochka extended the Dirac’ theorem on the minimum number of edges in critical graphs to the Dirac’s theorem on the minimum number of edges in DP-critical graphs, yielding a solution to the problem posed by Kostochka and Stiebitz [19].

A graph $G$ is DP-degree-colorable if $(H,f)$ has a DP-coloring whenever $f$ is a function from $V(H)$ to $\{0,1\}$ and $f(v,1)+f(v,2)+\dots+f(v,\kappa)\geq\deg_{G}(v)$ for each $v\in V(G)$. A GDP-tree is a connected graph in which every block is either a cycle or a complete graph. Bernshteyn, Kostochka, and Pron [6] gave a Brooks’ type result for DP-coloring. More detailed characterization of DP-degree-colorable multigraphs can be found in [17].

Theorem 1.1** (Bernshteyn, Kostochka and Pron [6]).**

Let $G$ be a connected graph. The graph $G$ is not DP-degree-colorable if and only if $G$ is a GDP-tree.

Dvořák and Postle [13] showed that every planar graph is DP-$5$-colorable, and observed that $\chi_{\mathrm{DP}}(G)\leq\kappa+1$ if $G$ is $\kappa$-degenerate. Thomassen [25] showed that every planar graph is $5$-choosable, and Voigt [26] showed that there are planar graphs which are not $4$-choosable. Thus it is interesting to give sufficient conditions for planar graphs to be $4$-choosable. As a generalization of list coloring, it is also interesting to give sufficient conditions for planar graphs to be DP-$4$-colorable. Kim and Ozeki [16] showed that for each $\kappa\in\{3,4,5,6\}$, every planar graph without $\kappa$-cycles is DP-$4$-colorable. Two cycles are adjacent if they have at least one edge in common. Kim and Yu [18] showed that every planar graph without triangles adjacent to $4$-cycles is DP-$4$-colorable. Some other materials on DP-coloring, see [5, 3, 1, 7, 2].

Let $G$ be a graph and $(H,f)$ be a valued cover of $G$. The pair $(H,f)$ is minimal non-strictly $f$-degenerate if $H$ has no strictly $f$-degenerate transversal, but $(H-L_{v},f)$ has a strictly $f$-degenerate transversal for any $v\in V(G)$. A toroidal graph is a graph that can be embedded in a torus. Any graph which can be embedded in a plane can also be embedded in a torus, thus every planar graph is also a toroidal graph. A $\kappa$-cap is a chordless cycle together with a vertex which is adjacent to exactly two adjacent vertices on the cycle. A $\kappa$-cap-subgraph $F_{\kappa}$ of a graph $G$ is a subgraph isomorphic to the $\kappa$-cap and all the vertices having degree four in $G$, see an example of $F_{5}$ in Fig. 1.

In Section 3, we show the following structural result on certain toroidal graphs.

Theorem 1.2**.**

Every connected toroidal graph without subgraphs isomorphic to the configurations in Fig. 2 has minimum degree at most three, unless it is a 2-connected $4$-regular graph with Euler characteristic $\epsilon(G)=0$.

Theorem 1.3**.**

Let $G$ be a planar graph without subgraphs isomorphic to the configurations in Fig. 2, and let $(H,f)$ be a valued cover of $G$. If $f(v,1)+f(v,2)+\dots+f(v,\kappa)\geq 4$ for each $v\in V(G)$, then $H$ has a strictly $f$-degenerate transversal.

Theorem 1.4**.**

Let $G$ be a toroidal graph without subgraphs isomorphic to the configurations in Fig. 2, and let $(H,f)$ be a valued cover of $G$. If $f(v,1)+f(v,2)+\dots+f(v,\kappa)\geq 4$ for each $v\in V(G)$, and $(H_{S},f)$ is not a monoblock whenever $G[S]$ is a 2-connected $4$-regular graph, then $H$ has a strictly $f$-degenerate transversal.

The following structural result on toroidal graphs without $4$-cycles can be derived from the proof of Theorem 1.9 in [12]. For completeness and readable, we present a proof in Section 3.

Theorem 1.5** (Choi and Zhang [12]).**

If $G$ is a connected toroidal graph without $4$-cycles, then $G$ has minimum degree at most three or $G$ contains a $\kappa$-cap-subgraph for some $\kappa\geq 5$.

Theorem 1.6**.**

Let $G$ be a toroidal graph without $4$-cycles. Let $H$ be a cover of $G$ and $f$ be a function from $V(H)$ to $\{0,1,2\}$. If $f(v,1)+f(v,2)+\dots+f(v,\kappa)\geq 4$ for each $v\in V(G)$, then $H$ has a strictly $f$-degenerate transversal.

In Section 4, we show that every planar graph without subgraphs isomorphic to the configurations in Fig. 3 has minimum degree at most three unless it contains a $5$-cap-subgraph.

Theorem 1.7**.**

If $G$ is a planar graph without subgraphs isomorphic to the configurations in Fig. 3, then it has minimum degree at most three or it contains a $5$-cap-subgraph.

Theorem 1.8**.**

Let $G$ be a planar graph without subgraphs isomorphic to the configurations in Fig. 3. Let $H$ be a cover of $G$ and $f$ be a function from $V(H)$ to $\{0,1,2\}$. If $f(v,1)+f(v,2)+\dots+f(v,\kappa)\geq 4$ for each $v\in V(G)$, then $H$ has a strictly $f$-degenerate transversal.

2 Preliminary

We need three classes of graphs as the following.

• •

The graph $\widetilde{K_{p}}$ is the Cartesian product of the complete graph $K_{p}$ and an independent $\kappa$-set.

• •

The circular ladder graph $\Gamma_{n}$ is the Cartesian product of the cycle $C_{n}$ and an independent set with two vertices.

• •

The Möbius ladder $M_{n}$ is the graph with vertex set \big{\{}\,(i,j)\mid i\in[n],j\in[2]\,\big{\}}, in which two vertices $(i,j)$ and $(i^{\prime},j^{\prime})$ are adjacent if and only if either

1. —

$i^{\prime}=i+1$ and $j=j^{\prime}$ for $1\leq i\leq n-1$, or 2. —

$i=n$, $i^{\prime}=1$ and $j\neq j^{\prime}$.

Definition 1**.**

Let $(H,f)$ be a valued cover of a graph $G$. A kernel of $H$ is the subgraph obtained from $H$ by deleting each vertex $(u,j)$ with $f(u,j)=0$.

Definition 2**.**

Let $B$ be a graph. A building cover is a valued cover $(H,f)$ of $B$ such that

[TABLE]

for each $v\in V(B)$ and at least one of the following holds:

1. (i)

The kernel of $H$ is isomorphic to $B$. We call this cover a monoblock. 2. (ii)

If $B$ is isomorphic to a complete graph $K_{p}$ for some $p\geq 2$, then the kernel of $H$ is isomorphic to $\widetilde{K_{p}}$ with $f$ being constant on each component of $\widetilde{K_{p}}$. 3. (iii)

If $B$ is isomorphic to an odd cycle, then the kernel of $H$ is isomorphic to the circular ladder graph with $f\equiv 1$. 4. (iv)

If $B$ is isomorphic to an even cycle, then the kernel of $H$ is isomorphic to the Möbius ladder with $f\equiv 1$. ∎

Definition 3**.**

Every building cover is constructible. A valued cover $(H,f)$ of a graph $G$ is also constructible if it is obtained from a constructible valued cover $(H^{(1)},f^{(1)})$ of $G^{(1)}$ and a constructible valued cover $(H^{(2)},f^{(2)})$ of $G^{(2)}$ such that all of the following hold:

1. (i)

the graph $G$ is obtained from $G^{(1)}$ and $G^{(2)}$ by identifying $w_{1}$ in $G^{(1)}$ and $w_{2}$ in $G^{(2)}$ into a new vertex $w$, and 2. (ii)

the cover $H$ is obtained from $H^{(1)}$ and $H^{(2)}$ by identifying $(w_{1},q)$ and $(w_{2},q)$ into a new vertex $(w,q)$ for each $q\in[\kappa]$, and 3. (iii)

$f(w,q)=f^{(1)}(w_{1},q)+f^{(2)}(w_{2},q)$ for each $q\in[\kappa]$, $f=f^{(1)}$ on $H^{(1)}-L_{w}$, and $f=f^{(2)}$ on $H^{(2)}-L_{w}$. We simply write $\bm{f=f^{(1)}+f^{(2)}}$. ∎

Theorem 2.1** (Lu, Wang and Wang [21]).**

Let $G$ be a connected graph and $(H,f)$ be a valued cover with $f(v,1)+f(v,2)+\dots+f(v,\kappa)\geq\deg_{G}(v)$ for each vertex $v\in V(G)$. Thus $H$ has a strictly $f$-degenerate transversal if and only if $H$ is non-constructible.

Let $\mathcal{D}$ be the set of all the vertices $v\in V(G)$ such that $f(v,1)+f(v,2)+\dots+f(v,\kappa)\geq\deg_{G}(v)$.

Theorem 2.2** (Lu, Wang and Wang [21]).**

Let $G$ be a graph and $(H,f)$ be a valued cover of $G$. Let $F$ be a $2$-connected subgraph of $G$ with $V(F)\subseteq\mathcal{D}$. If $(H,f)$ is a minimal non-strictly $f$-degenerate pair, then

1. (i)

$G$* is connected and $f(v,1)+f(v,2)+\dots+f(v,\kappa)\leq\deg_{G}(v)$ for each $v\in V(G)$, and* 2. (ii)

$G^{\prime}=G[V(F)]$* is a cycle or a complete graph or $\deg_{G^{\prime}}(v)\leq\max_{q}f(v,q)$ for each $v\in V(G^{\prime})$.*

3 Certain toroidal graphs

We recall our structural result on some toroidal graphs. See 1.2

• Proof.

Suppose that $G$ is a connected toroidal graph without subgraphs isomorphic to that in Fig. 2 and the minimum degree is at least four. We may assume that $G$ has been 2-cell embedded in the plane or torus.

We give the initial charge $\mu(v)=\deg(v)-4$ for any $v\in V(G)$ and $\mu(f)=\deg(f)-4$ for any $f\in F(G)$. By Euler’s formula, the sum of the initial charges is $-4\epsilon(G)$. That is,

[TABLE]

1. R1

If a $3$-face is adjacent to three $5^{+}$-faces, then it receives $\frac{1}{3}$ from each adjacent face; 2. R2

Let $w_{1}w_{2}$ be incident to two $3$-faces $f=w_{1}w_{2}w_{3}$ and $w_{1}w_{2}w_{4}$. If $\deg(w_{1})=\deg(w_{2})=4$, then $f$ receives $\frac{1}{2}$ from each adjacent $5^{+}$-face; otherwise $f$ receives $\frac{1}{3}$ from each adjacent $5^{+}$-face and $\frac{1}{3}$ from each $5^{+}$-vertex in $\{w_{1},w_{2}\}$.

Since each $5$-cycle has no chords, each $3$-face is adjacent to at most one $3$-face and at least two $5^{+}$-faces. If a $3$-face $f$ is adjacent to three $5^{+}$-faces, then it receives $\frac{1}{3}$ from each adjacent $5^{+}$-face, and then $\mu^{\prime}(f)=3-4+3\times\frac{1}{3}=0$ by R1. If a $3$-face $f$ is adjacent to one $3$-face and two $5^{+}$-faces, then $\mu^{\prime}(f)\geq 3-4+2\times\frac{1}{2}=0$ or $\mu^{\prime}(f)\geq 3-4+3\times\frac{1}{3}=0$ by R2. Each $4$-face is not involved in the discharging procedure, so its final charge is zero. By the absence of the configuration in 2(b), each $5$-face $f$ is adjacent to at most two $3$-faces, thus it sends at most $\frac{1}{2}$ to each adjacent $3$-face and $\mu^{\prime}(f)\geq 5-4-2\times\frac{1}{2}=0$.

Let $w_{1}w_{2}$ be incident to two $3$-faces $w_{1}w_{2}w_{3}$ and $w_{1}w_{2}w_{4}$, and let $w_{2}w_{3}$ be incident to a $6^{+}$-face $f=ww_{2}w_{3}\dots w$. If $\deg(w_{1})=\deg(w_{2})=4$, then $ww_{2}$ is incident to $f$ and another $4^{+}$-face, and then $f$ sends $\frac{1}{2}$ to the $3$-face $w_{1}w_{2}w_{3}$, but sends nothing to the $4^{+}$-face incident to $ww_{2}$. In this case, we can view it as $f$ directly sends $\frac{1}{3}$ to $w_{1}w_{2}w_{3}$ and sends $\frac{1}{6}$ via $ww_{2}$ to $w_{1}w_{2}w_{3}$. Hence, $f$ averagely sends at most $\frac{1}{3}=2\times\frac{1}{6}$ to each adjacent face, thus each $6^{+}$-face $f$ has final charge $\mu^{\prime}(f)\geq\deg(f)-4-\deg(f)\times\frac{1}{3}\geq 0$. In particular, every $7^{+}$-face has positive final charge.

Each $4$-vertex is not involved in the discharging procedure, so its final charge is zero. Since there are no three consecutive $3$-faces, every $\kappa$-vertex is incident to at most $\frac{2\kappa}{3}$ triangular faces. By R2, each $5^{+}$-vertex $v$ sends at most $2\times\frac{1}{3}=\frac{2}{3}$ to two adjacent $3$-faces but sends nothing to singular $3$-face, thus

[TABLE]

By (1), every element in $V(G)\cup F(G)$ has final charge zero, thus $G$ is $4$-regular, $\epsilon(G)=0$ and $G$ has only $6^{-}$-faces. If $w$ is incident to a cut-edge $ww_{1}$, then $ww_{1}$ is incident to an $8^{+}$-face, a contradiction.

Suppose that $w$ is a cut-vertex but it is not incident to any cut-edge. Note that $G$ is $4$-regular, it follows that $G-w$ has exactly two components $C_{1}$ and $C_{2}$, and $w$ has exactly two neighbors in each of $C_{1}$ and $C_{2}$. We may assume that $ww_{1},ww_{2},ww_{3}$ and $ww_{4}$ are four incident edges in a cyclic order and $w_{1},w_{2}\in V(C_{1})$ and $w_{3},w_{4}\in V(C_{2})$. It is observed that the face $D$ incident to $ww_{2}$ and $ww_{3}$ is also incident to $ww_{1}$ and $ww_{4}$, thus $D$ must be a $6$-face with boundary $ww_{1}w_{2}ww_{3}w_{4}w$. Since $G$ is a simple $4$-regular graph, none of $ww_{1},ww_{2},ww_{3}$ and $ww_{4}$ is incident to a $3$-face, which implies that $\mu^{\prime}(D)\geq\mu(D)-2\times\frac{1}{2}=1>0$, a contradiction. Hence, $G$ has no cut-vertex and it is a $2$-connected $4$-regular graph. ∎

Corollary 1** (Cai, Wang and Zhu [10]).**

Every connected toroidal graph without $5$-cycles has minimum degree at most three unless it is a $4$-regular graph.

The following corollary is a direct consequence of Theorem 1.2, which is stronger than that every planar graph without $5$-cycles is $3$-degenerate [27].

Corollary 2**.**

Every planar graph without subgraphs isomorphic to that in Fig. 2 is 3-degenerate.

See 1.3

• Proof.

Suppose that $G$ is a counterexample to Theorem 1.3 with minimum number of vertices. It is observed that $G$ is connected and $(H,f)$ is a minimal non-strictly $f$-degenerate pair. By Corollary 2, the minimum degree of $G$ is at most three, but this contradicts Theorem 2.2(i). ∎

Remark 1**.**

Note that not every toroidal graph without subgraphs isomorphic to that in Fig. 2 is $3$-degenerate. For example, the Cartesian product of an $m$-cycle and an $n$-cycle is a 2-connected 4-regular graph with Euler characteristic $\epsilon(G)=0$.

We recall our main result on some toroidal graphs. See 1.4

• Proof.

Suppose that $G$ is a counterexample to Theorem 1.4 with minimum number of vertices. It is observed that $G$ is connected and $(H,f)$ is a minimal non-strictly $f$-degenerate pair. By Theorem 1.2, the minimum degree of $G$ is at most three or it is a 2-connected $4$-regular graph with Euler characteristic $\epsilon(G)=0$. By Theorem 2.2(i), the minimum degree of $G$ is at least four, which implies that $G$ is a 2-connected $4$-regular graph with Euler characteristic $\epsilon(G)=0$. Here, we have that

[TABLE]

Note that $G$ is neither a cycle nor a $4$-regular complete graph. On the other hand, $(H,f)$ is not a monoblock, which contradicts Theorem 2.1. ∎

Corollary 3** (Cai, Wang and Zhu [10]).**

(i) Every toroidal graph without $3$-cycles is $4$-choosable. (ii) Every toroidal graph without $5$-cycles is $4$-choosable.

Recall the structural result on toroidal graphs without $4$-cycles. See 1.5

• Proof.

Suppose that $G$ is a connected toroidal graph and it has the properties: (1) it has no $4$-cycles; (2) the minimum degree is at least four; and (3) it contains no $\kappa$-cap-subgraphs for any $\kappa\geq 5$. Since $G$ has no $4$-cycles, there are no $3$-cap-subgraphs or $4$-cap-subgraphs. We may assume that $G$ has been 2-cell embedded in the plane or torus.

A vertex is bad if it is a vertex of degree $4$ and it is incident to two $3$-faces; a vertex is good if it is not bad. A $3$-face is called a bad $3$-face if it is incident to a bad vertex. Let $H=H(G)$ be the graph where $V(H)$ is the set of $3$-faces of $G$ incident to at least one bad vertex and $uv\in E(H)$ if and only if the two $3$-faces that correspond to $u$ and $v$ has a common bad vertex.

1. ($\ast$)

The graph $H(G)$ has maximum degree at most three. Every component of $H(G)$ is a cycle or a tree.

• Proof.

Note that each bad vertex of $G$ corresponds to an edge in $H(G)$, thus each component of $H(G)$ has at least one edge. Since each $3$-face is incident to at most three bad vertices, the maximum degree of $H(G)$ is at most three. By the absence of $\kappa$-cap-subgraphs, there is no vertex $v$ in $H(G)$ such that it has three neighbors and it is contained in a cycle. Hence, each component of $H(G)$ is a cycle or a tree. ∎

We assign each vertex $v\in V(G)$ an initial charge $\mu(v)=\deg(v)-6$ and each face $f\in F(G)$ an initial charge $\mu(f)=2\deg(f)-6$. According to Euler’s formula,

[TABLE]

Next, we design a discharging procedure to get a final charge $\mu^{\prime}(x)$ for each $x\in V(G)\cup F(G)$. For the discharging part, we introduce a notion bank. For each component $D$ of $H(G)$, we give it a bank $B(D)$ which has an initial charge zero. A good vertex is incident to a bank $B(D)$ if it is incident to a bad $3$-face $f$ and $f$ is a vertex in the component $D$.

1. R1

Each face distributes its initial charge uniformly to each incident vertex. 2. R2

Each good vertex $v$ sends $\frac{2}{5}$ to each incident bank via each incident bad $3$-face. 3. R3

For each component $D$ of $H(G)$, the bank $B(D)$ sends $\frac{2}{5}$ to each bad vertex in $G$ that corresponds to an edge in $D$.

Each face $f$ with $\deg(f)\geq 5$ sends charge $\frac{\mu(f)}{\deg(f)}=\frac{2\deg(f)-6}{\deg(f)}\geq\frac{4}{5}$ to each incident vertex. Since $G$ has no $4$-cycles, there are neither $4$-faces nor adjacent $3$-faces. This implies that each vertex $v$ is incident to at most $\frac{\deg(v)}{2}$ banks.

Let $v$ be a vertex. If $\deg(v)\geq 5$, then $v$ receives at least $\frac{4}{5}\times\lceil\frac{\deg(v)}{2}\rceil$ from its incident $5^{+}$-faces and sends at most $\frac{2}{5}\times\lfloor\frac{\deg(v)}{2}\rfloor$ to its incident banks, which implies that $\mu^{\prime}(v)\geq\deg(v)-6+\frac{4}{5}\times\lceil\frac{\deg(v)}{2}\rceil-\frac{2}{5}\times\lfloor\frac{\deg(v)}{2}\rfloor>0$. If $v$ is a bad $4$-vertex, then it receives $\frac{4}{5}$ from each incident $5^{+}$-face and $\frac{2}{5}$ from its incident bank, which implies that $\mu^{\prime}(v)\geq 4-6+2\times\frac{4}{5}+\frac{2}{5}=0$. If $v$ is a good $4$-vertex, then it is incident to at most one $3$-face, and then it receives $\frac{4}{5}$ from each incident $5^{+}$-face and sends at most $\frac{2}{5}$ to its incident bank, which implies that $\mu^{\prime}(v)\geq 4-6+3\times\frac{4}{5}-\frac{2}{5}=0$.

Let $D$ be a component of $H(G)$ and $D$ be a cycle with $n$ vertices. Since $D$ is a cycle, each $3$-face in $G$ that corresponds to a vertex in $D$ must be incident to a good vertex, and each such good vertex sends $\frac{2}{5}$ to each incident bank via each incident bad $3$-face. Thus, the final charge of the bank $B(D)$ is $\frac{2}{5}n-\frac{2}{5}n=0$.

Let $D$ be a component of $H(G)$ and $D$ is a tree with $n$ vertices. For each $i$ in $\{1,2,3\}$, let $\alpha_{i}$ be the number of vertices of degree $i$ in $D$. Each $3$-face in $G$ that corresponds to a vertex of degree one is incident to two good vertices and each $3$-face in $G$ that corresponds to a vertex of degree two is incident to exactly one good vertex. Thus the bank $B(D)$ receives $\frac{2}{5}\times 2\alpha_{1}+\frac{2}{5}\times\alpha_{2}=\frac{4}{5}\alpha_{1}+\frac{2}{5}\alpha_{2}$, and sends $\frac{2}{5}|E(D)|=\frac{2}{5}(n-1)$. Since $n=\alpha_{1}+\alpha_{2}+\alpha_{3}$ and $2(n-1)=\alpha_{1}+2\alpha_{2}+3\alpha_{3}$, we have that $2\alpha_{1}+\alpha_{2}=n+2$. Hence, the final charge of the bank $B(D)$ is $\frac{4}{5}\alpha_{1}+\frac{2}{5}\alpha_{2}-\frac{2}{5}(n-1)=\frac{6}{5}>0$.

The discharging procedure preserves the total charge, thus the sum of the final charge should be zero by (2). This implies that $G$ is $4$-regular and no component of $H(G)$ is a tree. By the absence of $\kappa$-cap-subgraphs for any $\kappa\geq 3$, no component of $H(G)$ is a cycle. Hence, $H(G)$ does not exist and every vertex in $G$ is good. For every vertex $v$ (having degree four), the final charge is $4-6+3\times\frac{4}{5}>0$, a contradiction. ∎

See 1.6

• Proof.

Suppose that $G$ is a counterexample to Theorem 1.6 with minimum number of vertices. It is observed that $G$ is connected and $(H,f)$ is a minimal non-strictly $f$-degenerate pair. By Theorem 1.5 and Theorem 2.2(i), $G$ must contain a $\kappa$-cap-subgraph $F$ for some $\kappa\geq 5$. Note that

[TABLE]

Moreover, $G[V(F)]$ is $2$-connected and it is neither a cycle nor a complete graph. On the other hand, there exists a vertex $v\in V(F)$ such that $\deg_{G[V(F)]}(v)\geq 3>\max_{q}f(v,q)$, which contradicts Theorem 2.2(ii). ∎

4 Certain planar graphs

Lam, Xu and Liu [20] showed that every planar graph without $4$-cycles has minimum degree at most three unless it contains a $5$-cap-subgraph. Borodin and Ivanova [9] further improved this by showing every planar graph without triangles adjacent to $4$-cycles has minimum degree at most three unless it contains a $5$-cap-subgraph. Kim and Yu [18] recovered this structure and showed that every planar graph without triangles adjacent to $4$-cycles is DP-$4$-colorable.

Borodin and Ivanova [9] (independently, Cheng–Chen–Wang [11]) showed that every planar graph without triangles adjacent to $4$-cycle is $4$-choosable. Xu and Wu [28] showed that a planar graph without $5$-cycles simultaneously adjacent to $3$-cycles and $4$-cycles is $4$-choosable. Actually, they gave the following stronger structural result.

Theorem 4.1** (Xu and Wu [28]).**

If $G$ is a planar graph without subgraphs isomorphic to the configurations in Fig. 4, then it has minimum degree at most three unless it contains a $5$-cap-subgraph.

We recall the structural result on some planar graphs, which improves Theorem 4.1. See 1.7

• Proof.

Suppose that $G$ is a counterexample to Theorem 1.7. We may assume that it is connected and it has been $2$-cell embedded in the plane. Since each $5$-cycle has no chords, each $3$-face is adjacent to at most one $3$-face and at least two $5^{+}$-faces.

We define an initial charge function by setting $\mu(v)=\deg(v)-4$ for each $v\in V(G)$ and $\mu(f)=\deg(f)-4$ for each $f\in F(G)$. By Euler’s formula, the sum of the initial charges is $-8$. That is,

[TABLE]

An $(a,b)$-edge is an edge with endpoints having degree $a$ and $b$. Let $f$ be a $5$-face incident to at least one $5^{+}$-vertex. If $f$ is adjacent to five $3$-faces, then we call $f$ a $\bm{5_{\mathrm{a}}}$-face. If $f$ is adjacent to exactly four $3$-faces, then we call $f$ a $\bm{5_{\mathrm{b}}}$-face. If $f$ is adjacent to three $3$-faces and one of the $3$-faces is adjacent to another $3$-face via a $(4,4)$-edge, then we call $f$ a $\bm{5_{\mathrm{c}}}$-face. Note that the $5_{\mathrm{c}}$-face can only be as illustrated in 5(c).

If $f=w_{1}w_{2}w_{3}w_{4}w_{5}$ is a $5$-face incident to five $4$-vertices and adjacent to a $3$-face $ww_{1}w_{2}$ and $w$ is a $5^{+}$-vertex, then we call $w$ a related source of $f$ and $f$ a sink of $w$.

1. R1

If a $3$-face is adjacent to three $5^{+}$-faces, then it receives $\frac{1}{3}$ from each adjacent face. 2. R2

Let $w_{1}w_{2}$ be incident to two $3$-faces $f=w_{1}w_{2}w_{3}$ and $w_{1}w_{2}w_{4}$. If $\deg(w_{1})=\deg(w_{2})=4$, then $f$ receives $\frac{1}{2}$ from each adjacent $5^{+}$-face; otherwise $f$ receives $\frac{1}{3}$ from each adjacent $5^{+}$-face and $\frac{1}{3}$ from each $5^{+}$-vertex in $\{w_{1},w_{2}\}$. 3. R3

If $f$ is a $5$-face incident to five $4$-vertices and adjacent to at least four $3$-faces, then $f$ receives $\frac{1}{6}$ from each of its related source via the adjacent $3$-face. 4. R4

If $f$ is a $5_{\mathrm{a}}$-face, then it receives $\frac{2}{3\#}$ from each incident $5^{+}$-vertex, where $\#$ is the number of incident $5^{+}$-vertices. 5. R5

If $f$ is a $5_{\mathrm{b}}$-face, then it receives $\frac{1}{3\#}$ from each incident $5^{+}$-vertex, where $\#$ is the number of incident $5^{+}$-vertices. 6. R6

If $f$ is a $5_{\mathrm{c}}$-face, then it receives $\frac{1}{6}$ from each incident $5^{+}$-vertex.

The final charge of each face is nonnegative. If a $3$-face $f$ is adjacent to three $5^{+}$-faces, then it receives $\frac{1}{3}$ from each adjacent $5^{+}$-face, and then $\mu^{\prime}(f)=3-4+3\times\frac{1}{3}=0$ by R1. If a $3$-face $f$ is adjacent to one $3$-face and two $5^{+}$-faces, then $\mu^{\prime}(f)\geq 3-4+2\times\frac{1}{2}=0$ or $\mu^{\prime}(f)\geq 3-4+3\times\frac{1}{3}=0$ by R2. Each $4$-face is not involved in the discharging procedure, so its final charge is zero.

Let $f$ be a $5$-face incident to five 4-vertices and $\alpha$ be the number of adjacent $3$-faces. By the absence of $5$-cap-subgraphs, $f$ sends $\frac{1}{3}$ to each adjacent $3$-face by R1 and R2. If $\alpha\leq 3$, then $\mu^{\prime}(f)=5-4-\alpha\times\frac{1}{3}\geq 0$. If $\alpha\geq 4$, then $f$ receives $\frac{1}{6}$ from each related source, which implies that $\mu^{\prime}(f)=5-4-\alpha\times\frac{1}{3}+\alpha\times\frac{1}{6}>0$ by R1 and R3. So we may assume that $f$ is incident to at least one $5^{+}$-vertex.

If $f$ is a $5_{\mathrm{a}}$-face, then it sends $\frac{1}{3}$ to each adjacent $3$-face, and then $\mu^{\prime}(f)=5-4-5\times\frac{1}{3}+\#\times\frac{2}{3\#}=0$ by R1 and R4. If $f$ is a $5_{\mathrm{b}}$-face, then it sends $\frac{1}{3}$ to each adjacent $3$-face, and then $\mu^{\prime}(f)=5-4-4\times\frac{1}{3}+\#\times\frac{1}{3\#}=0$ by R1 and R5. If $f$ is a $5_{\mathrm{c}}$-face, then it sends $\frac{1}{2}$ to an adjacent $3$-face and sends $\frac{1}{3}$ to each of the other adjacent $3$-face, and then $\mu^{\prime}(f)\geq 5-4-\frac{1}{2}-2\times\frac{1}{3}+\frac{1}{6}=0$ by R1, R2 and R6. If $f$ is incident to exactly three $3$-faces but it is not a $5_{\mathrm{c}}$-face, then it sends $\frac{1}{3}$ to each adjacent $3$-face, and $\mu^{\prime}(f)=5-4-3\times\frac{1}{3}=0$ by R1 and R2. If $f$ is incident to at most two $3$-faces, then it sends at most $\frac{1}{2}$ to each adjacent $3$-face, and $\mu^{\prime}(f)\geq 5-4-2\times\frac{1}{2}=0$ by R1 and R2.

By the same argument as in Theorem 1.2, every $6^{+}$-face averagely sends at most $\frac{1}{3}$ via each incident edge, thus each $6^{+}$-face $f$ has final charge $\mu^{\prime}(f)\geq\deg(f)-4-\deg(f)\times\frac{1}{3}\geq 0$. In particular, each $7^{+}$-face has positive final charge.

The final charge of each vertex is nonnegative. Each $4$-vertex is not involved in the discharging procedure, so its final charge is zero. Let $v$ be a $6^{+}$-vertex incident with a face $f$. If $f$ is neither a $3$-face nor a $5$-face, then $v$ sends nothing to $f$. If $f$ is a $3$-face, then $v$ possibly sends $\frac{1}{3}$ to $f$ by R2, or sends $\frac{1}{6}$ via $f$ to a sink by R3. If $f$ is a $5$-face other than $5_{\mathrm{a}}$-face incident to exactly one $5^{+}$-vertex, then $v$ sends at most $\frac{1}{3}$ to $f$. If $f$ is a $5_{\mathrm{a}}$-face incident to exactly one $5^{+}$-vertex, then $v$ sends $\frac{2}{3}$ to $f$, but it sends nothing to/via the two $3$-faces adjacent to $f$, otherwise there is a $5$-cap-subgraph. Hence, $v$ averagely sends at most $\frac{1}{3}$ to/via incident face in any case, thus $\mu^{\prime}(v)\geq\deg(v)-4-\deg(v)\times\frac{1}{3}\geq 0$.

Let $v$ be a $5$-vertex incident to five consecutive faces $f_{1},f_{2},f_{3},f_{4}$ and $f_{5}$. We divide the discussions into four cases.

(i) Suppose that $v$ is incident to two adjacent $3$-faces, say $f_{1}$ and $f_{2}$. Since each $5$-cycle has no chords, neither $f_{3}$ nor $f_{5}$ is a $4^{-}$-face. If $f_{3}$ is a $5$-face, then it is adjacent to at most three $3$-faces by the absence of 3(b) and 3(c), but it is not a $5_{\mathrm{c}}$ -face, and then it receives nothing from $v$. If $f_{3}$ is a $6^{+}$-face, then it also receives nothing from $v$. To sum up, $v$ sends nothing to $f_{3}$, and symmetrically sends nothing to $f_{5}$. By the discharging rules, if $f_{4}$ is a $4^{+}$-face, then $v$ sends nothing to $f_{4}$; while $f_{4}$ is a $3$-face, then $v$ possibly sends $\frac{1}{6}$ via $f_{4}$ to a sink. Note that $v$ sends out nothing via $f_{1}$ or $f_{2}$ by the absence of 3(b) and 3(c). Hence, $\mu^{\prime}(v)\geq 5-4-2\times\frac{1}{3}-\frac{1}{6}>0$.

(ii) Suppose that $v$ is incident to two nonadjacent $3$-faces, say $f_{1}$ and $f_{3}$. If the common edge between $f_{4}$ and $f_{5}$ is $(5,5^{+})$-edge, then $v$ sends at most $\frac{1}{6}$ to each of $f_{4}$ and $f_{5}$ by R5 and R6, and then $v$ sends at most $\frac{1}{3}$ in total to $f_{4}$ and $f_{5}$. If the common edge between $f_{4}$ and $f_{5}$ is a $(5,4)$-edge, then $v$ sends $\frac{1}{3}$ to one face in $\{f_{4},f_{5}\}$ and sends nothing to the other face, or $v$ sends at most $\frac{1}{6}$ to each of $f_{4}$ and $f_{5}$, thus $v$ sends at most $\frac{1}{3}$ in total to $f_{4}$ and $f_{5}$. To sum up, $v$ sends at most $\frac{1}{3}$ in total to $f_{4}$ and $f_{5}$ in any case. If $v$ sends $\frac{2}{3}$ to $f_{2}$, then $f_{2}$ must be a $5_{\mathrm{a}}$-face incident to exactly one $5^{+}$-vertex and $v$ cannot be a related source of some faces, which implies that $\mu^{\prime}(v)\geq 5-4-\frac{2}{3}-\frac{1}{3}=0$. If $f$ sends at most $\frac{1}{3}$ to $f_{2}$, then $v$ could be related sources of two $5$-faces, and then $\mu^{\prime}(v)\geq 5-4-2\times\frac{1}{3}-2\times\frac{1}{6}=0$.

(iii) Suppose that $v$ is incident to exactly one $3$-face $f_{1}$. By the discharging rules, $v$ sends at most $\frac{1}{3}$ to each of $f_{2}$ and $f_{5}$, nothing to each of $f_{3}$ and $f_{4}$, and possibly $\frac{1}{6}$ via $f_{1}$ to a sink, which implies that $\mu^{\prime}(v)\geq 5-4-2\times\frac{1}{3}-\frac{1}{6}>0$.

(iv) If $v$ is not incident to any $3$-face, then $\mu^{\prime}(v)=\mu(v)=5-4=1>0$. ∎

We recall our main result on certain planar graphs. See 1.8

• Proof.

Suppose that $G$ is a counterexample to Theorem 1.8 with minimum number of vertices. It is observed that $G$ is connected and $(H,f)$ is a minimal non-strictly $f$-degenerate pair. By Theorem 1.7 and Theorem 2.2(i), $G$ must contain a $5$-cap-subgraph $F_{5}$. Note that

[TABLE]

Moreover, $G[V(F_{5})]$ is $2$-connected and it is neither a cycle nor a complete graph. On the other hand, there exists a vertex $v\in V(F_{5})$ such that $\deg_{G[V(F_{5})]}(v)\geq 3>\max_{q}f(v,q)$, which contradicts Theorem 2.2(ii). ∎

Remark 2**.**

Each of the graph in Fig. 3 contains a $3$-cycle, a $4$-cycle and a $5$-cycle, and these three short cycles are mutually adjacent. Thus,

1. (i)

if $G$ is a planar graph without $3$-cycles adjacent to $4$-cycles, then it is DP-$4$-colorable and its list vertex arboricity is at most two; 2. (ii)

if $G$ is a planar graph without $3$-cycles adjacent to $5$-cycles, then it is DP-$4$-colorable and its list vertex arboricity is at most two; 3. (iii)

if $G$ is a planar graph without $4$-cycles adjacent to $5$-cycles, then it is DP-$4$-colorable and its list vertex arboricity is at most two.

Remark 3**.**

Theorem 1.7 cannot be extended to toroidal graphs; once again, the Cartesian product of an $m$-cycle and an $n$-cycle is a counterexample. Thus, it is interesting to extend Theorem 1.8 to toroidal graphs.

Kim and Ozeki [16] pointed out that DP-coloring is also a generalization of signed (list) coloring of a signed graph $(G,\sigma)$, thus Theorem 1.8 implies the following result, which partly extends that in [15, Theorem 3.5]. For details on signed (list) coloring of signed graph, we refer the reader to [24, 15, 22, 14].

Theorem 4.2**.**

If $(G,\sigma)$ be a signed planar graph and $G$ has no subgraphs isomorphic to the configurations in Fig. 3, then $(G,\sigma)$ is signed $4$-choosable.

Acknowledgments. This work was supported by the National Natural Science Foundation of China (xxxxxxxx) and partially supported by the Fundamental Research Funds for Universities in Henan (YQPY20140051).