# Variable degeneracy on toroidal graphs

Let f be a nonnegative integer valued function on the vertex-set of a graph. A graph is strictly f-degenerate if each nonempty subgraph Γ has a vertex v such that _Γ(v) < f(v). A cover of a graph G is a graph H with vertex set V(H) = _v ∈ V(G) L_v, where L_v = { (v, 1), (v, 2), ..., (v, κ) }; the edge set M = _uv ∈ E(G)M_uv, where M_uv is a matching between L_u and L_v. A vertex set R ⊆ V(H) is a transversal of H if |R ∩ L_v| = 1 for each v ∈ V(G). A transversal R is a strictly f-degenerate transversal if H[R] is strictly f-degenerate. In this paper, we give some structural results on planar and toroidal graphs with forbidden configurations, and give some sufficient conditions for the existence of strictly f-degenerate transversal by using these structural results.

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## 1 Introduction

All graphs considered in this paper are finite, undirected and simple. Let stand for the set of nonnegative integers, and let be a function from to . A graph is strictly -degenerate if each nonempty subgraph of has a vertex such that . A cover of a graph is a graph with vertex set , where ; the edge set , where is a matching between and . Note that is an independent set in and 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 is a transversal of if for each .

Let be a cover of and be a function from to , we call the pair a valued cover of . Let be a subset of , we use to denote the induced subgraph . A transversal is a strictly -degenerate transversal if is strictly -degenerate.

Let be a cover of and be a function from to . An independent transversal, or DP-coloring, of is a strictly -degenerate transversal of . It is observed that a DP-coloring is a special independent set in . DP-coloring, also known as correspondence coloring, is introduced by Dvořák and Postle [13]. It is shown [21] that strictly -degenerate transversal generalizes list coloring, -partition, signed coloring, DP-coloring and -forested-coloring.

The DP-chromatic number of is the least integer such that has a DP-coloring whenever is a cover of and is a function from to with for each . A graph is DP--colorable if its DP-chromatic number is at most .

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 to is -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 is DP-degree-colorable if has a DP-coloring whenever is a function from to and for each . 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 be a connected graph. The graph is not DP-degree-colorable if and only if is a GDP-tree.

Dvořák and Postle [13] showed that every planar graph is DP--colorable, and observed that if is -degenerate. Thomassen [25] showed that every planar graph is -choosable, and Voigt [26] showed that there are planar graphs which are not -choosable. Thus it is interesting to give sufficient conditions for planar graphs to be -choosable. As a generalization of list coloring, it is also interesting to give sufficient conditions for planar graphs to be DP--colorable. Kim and Ozeki [16] showed that for each , every planar graph without -cycles is DP--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 -cycles is DP--colorable. Some other materials on DP-coloring, see [5, 3, 1, 7, 2].

Let be a graph and be a valued cover of . The pair is minimal non-strictly -degenerate if has no strictly -degenerate transversal, but has a strictly -degenerate transversal for any . 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 -cap is a chordless cycle together with a vertex which is adjacent to exactly two adjacent vertices on the cycle. A -cap-subgraph of a graph is a subgraph isomorphic to the -cap and all the vertices having degree four in , see an example of 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 -regular graph with Euler characteristic .

###### Theorem 1.3.

Let be a planar graph without subgraphs isomorphic to the configurations in Fig. 2, and let be a valued cover of . If for each , then has a strictly -degenerate transversal.

###### Theorem 1.4.

Let be a toroidal graph without subgraphs isomorphic to the configurations in Fig. 2, and let be a valued cover of . If for each , and is not a monoblock whenever is a 2-connected -regular graph, then has a strictly -degenerate transversal.

The following structural result on toroidal graphs without -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 is a connected toroidal graph without -cycles, then has minimum degree at most three or contains a -cap-subgraph for some .

###### Theorem 1.6.

Let be a toroidal graph without -cycles. Let be a cover of and be a function from to . If for each , then has a strictly -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 -cap-subgraph.

###### Theorem 1.7.

If 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 -cap-subgraph.

###### Theorem 1.8.

Let be a planar graph without subgraphs isomorphic to the configurations in Fig. 3. Let be a cover of and be a function from to . If for each , then has a strictly -degenerate transversal.

## 2 Preliminary

We need three classes of graphs as the following.

• The graph is the Cartesian product of the complete graph and an independent -set.

• The circular ladder graph is the Cartesian product of the cycle and an independent set with two vertices.

• The Möbius ladder is the graph with vertex set , in which two vertices and are adjacent if and only if either

1. and for , or

2. , and .

###### Definition 1.

Let be a valued cover of a graph . A kernel of is the subgraph obtained from by deleting each vertex with .

###### Definition 2.

Let be a graph. A building cover is a valued cover of such that

 f(v,1)+f(v,2)+⋯+f(v,κ)=degB(v)

for each and at least one of the following holds:

1. The kernel of is isomorphic to . We call this cover a monoblock.

2. If is isomorphic to a complete graph for some , then the kernel of is isomorphic to with being constant on each component of .

3. If

is isomorphic to an odd cycle, then the kernel of

is isomorphic to the circular ladder graph with .

4. If is isomorphic to an even cycle, then the kernel of is isomorphic to the Möbius ladder with . ∎

###### Definition 3.

Every building cover is constructible. A valued cover of a graph is also constructible if it is obtained from a constructible valued cover of and a constructible valued cover of such that all of the following hold:

1. the graph is obtained from and by identifying in and in into a new vertex , and

2. the cover is obtained from and by identifying and into a new vertex for each , and

3. for each , on , and on . We simply write . ∎

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

Let be a connected graph and be a valued cover with for each vertex . Thus has a strictly -degenerate transversal if and only if is non-constructible.

Let be the set of all the vertices such that .

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

Let be a graph and be a valued cover of . Let be a -connected subgraph of with . If is a minimal non-strictly -degenerate pair, then

1. is connected and for each , and

2. is a cycle or a complete graph or for each .

## 3 Certain toroidal graphs

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

Suppose that 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 has been 2-cell embedded in the plane or torus.

We give the initial charge for any and for any . By Euler’s formula, the sum of the initial charges is . That is,

 ∑v∈V(G)(deg(v)−4)+∑f∈F(G)(deg(f)−4)=−4ϵ(G)≤0. (1)
1. If a -face is adjacent to three -faces, then it receives from each adjacent face;

2. Let be incident to two -faces and . If , then receives from each adjacent -face; otherwise receives from each adjacent -face and from each -vertex in .

Since each -cycle has no chords, each -face is adjacent to at most one -face and at least two -faces. If a -face is adjacent to three -faces, then it receives from each adjacent -face, and then by a. If a -face is adjacent to one -face and two -faces, then or by b. Each -face is not involved in the discharging procedure, so its final charge is zero. By the absence of the configuration in (b), each -face is adjacent to at most two -faces, thus it sends at most to each adjacent -face and .

Let be incident to two -faces and , and let be incident to a -face . If , then is incident to and another -face, and then sends to the -face , but sends nothing to the -face incident to . In this case, we can view it as directly sends to and sends via to . Hence, averagely sends at most to each adjacent face, thus each -face has final charge . In particular, every -face has positive final charge.

Each -vertex is not involved in the discharging procedure, so its final charge is zero. Since there are no three consecutive -faces, every -vertex is incident to at most triangular faces. By b, each -vertex sends at most to two adjacent -faces but sends nothing to singular -face, thus

 μ′(v)≥deg(v)−4−⌊deg(v)3⌋×23>0.

By (1), every element in has final charge zero, thus is -regular, and has only -faces. If is incident to a cut-edge , then is incident to an -face, a contradiction.

Suppose that is a cut-vertex but it is not incident to any cut-edge. Note that is -regular, it follows that has exactly two components and , and has exactly two neighbors in each of and . We may assume that and are four incident edges in a cyclic order and and . It is observed that the face incident to and is also incident to and , thus must be a -face with boundary . Since is a simple -regular graph, none of and is incident to a -face, which implies that , a contradiction. Hence, has no cut-vertex and it is a -connected -regular graph. ∎

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

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

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

###### Corollary 2.

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

See 1.3

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

###### Remark 1.

Note that not every toroidal graph without subgraphs isomorphic to that in Fig. 2 is -degenerate. For example, the Cartesian product of an -cycle and an -cycle is a 2-connected 4-regular graph with Euler characteristic .

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

Suppose that is a counterexample to Theorem 1.4 with minimum number of vertices. It is observed that is connected and is a minimal non-strictly -degenerate pair. By Theorem 1.2, the minimum degree of is at most three or it is a 2-connected -regular graph with Euler characteristic . By Theorem 2.2a, the minimum degree of is at least four, which implies that is a 2-connected -regular graph with Euler characteristic . Here, we have that

 V(G)=D={v∣f(v,1)+f(v,2)+⋯+f(v,κ)≥degG(v)=4}.

Note that is neither a cycle nor a -regular complete graph. On the other hand, is not a monoblock, which contradicts Theorem 2.1. ∎

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

(i) Every toroidal graph without -cycles is -choosable. (ii) Every toroidal graph without -cycles is -choosable.

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

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

A vertex is bad if it is a vertex of degree and it is incident to two -faces; a vertex is good if it is not bad. A -face is called a bad -face if it is incident to a bad vertex. Let be the graph where is the set of -faces of incident to at least one bad vertex and if and only if the two -faces that correspond to and has a common bad vertex.

1. The graph has maximum degree at most three. Every component of is a cycle or a tree.

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

We assign each vertex an initial charge and each face an initial charge . According to Euler’s formula,

 ∑v∈V(G)(deg(v)−6)+∑f∈F(G)(2deg(f)−6)=−6ϵ(G)≤0. (2)

Next, we design a discharging procedure to get a final charge for each . For the discharging part, we introduce a notion bank. For each component of , we give it a bank which has an initial charge zero. A good vertex is incident to a bank if it is incident to a bad -face and is a vertex in the component .

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

2. Each good vertex sends to each incident bank via each incident bad -face.

3. For each component of , the bank sends to each bad vertex in that corresponds to an edge in .

Each face with sends charge to each incident vertex. Since has no -cycles, there are neither -faces nor adjacent -faces. This implies that each vertex is incident to at most banks.

Let be a vertex. If , then receives at least from its incident -faces and sends at most to its incident banks, which implies that . If is a bad -vertex, then it receives from each incident -face and from its incident bank, which implies that . If is a good -vertex, then it is incident to at most one -face, and then it receives from each incident -face and sends at most to its incident bank, which implies that .

Let be a component of and be a cycle with vertices. Since is a cycle, each -face in that corresponds to a vertex in must be incident to a good vertex, and each such good vertex sends to each incident bank via each incident bad -face. Thus, the final charge of the bank is .

Let be a component of and is a tree with vertices. For each in , let be the number of vertices of degree in . Each -face in that corresponds to a vertex of degree one is incident to two good vertices and each -face in that corresponds to a vertex of degree two is incident to exactly one good vertex. Thus the bank receives , and sends . Since and , we have that . Hence, the final charge of the bank is .

The discharging procedure preserves the total charge, thus the sum of the final charge should be zero by (2). This implies that is -regular and no component of is a tree. By the absence of -cap-subgraphs for any , no component of is a cycle. Hence, does not exist and every vertex in is good. For every vertex (having degree four), the final charge is , a contradiction. ∎

See 1.6

Suppose that is a counterexample to Theorem 1.6 with minimum number of vertices. It is observed that is connected and is a minimal non-strictly -degenerate pair. By Theorem 1.5 and Theorem 2.2a, must contain a -cap-subgraph for some . Note that

 V(F)⊆D={v∣f(v,1)+f(v,2)+⋯+f(v,κ)≥degG(v)=4}.

Moreover, is -connected and it is neither a cycle nor a complete graph. On the other hand, there exists a vertex such that , which contradicts Theorem 2.2b. ∎

## 4 Certain planar graphs

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

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

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

If 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 -cap-subgraph.

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

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

We define an initial charge function by setting for each and for each . By Euler’s formula, the sum of the initial charges is . That is,

 ∑v∈V(G)(deg(v)−4)+∑f∈F(G)(deg(f)−4)=−8. (3)

An -edge is an edge with endpoints having degree and . Let be a -face incident to at least one -vertex. If is adjacent to five -faces, then we call a -face. If is adjacent to exactly four -faces, then we call a -face. If is adjacent to three -faces and one of the -faces is adjacent to another -face via a -edge, then we call a -face. Note that the -face can only be as illustrated in (c).

If is a -face incident to five -vertices and adjacent to a -face and is a -vertex, then we call a related source of and a sink of .

1. If a -face is adjacent to three -faces, then it receives from each adjacent face.

2. Let be incident to two -faces and . If , then receives from each adjacent -face; otherwise receives from each adjacent -face and from each -vertex in .

3. If is a -face incident to five -vertices and adjacent to at least four -faces, then receives from each of its related source via the adjacent -face.

4. If is a -face, then it receives from each incident -vertex, where is the number of incident -vertices.

5. If is a -face, then it receives from each incident -vertex, where is the number of incident -vertices.

6. If is a -face, then it receives from each incident -vertex.

The final charge of each face is nonnegative. If a -face is adjacent to three -faces, then it receives from each adjacent -face, and then by a. If a -face is adjacent to one -face and two -faces, then or by b. Each -face is not involved in the discharging procedure, so its final charge is zero.

Let be a -face incident to five 4-vertices and be the number of adjacent -faces. By the absence of -cap-subgraphs, sends to each adjacent -face by a and b. If , then . If , then receives from each related source, which implies that by a and c. So we may assume that is incident to at least one -vertex.

If is a -face, then it sends to each adjacent -face, and then by a and d. If is a -face, then it sends to each adjacent -face, and then by a and e. If is a -face, then it sends to an adjacent -face and sends to each of the other adjacent -face, and then by a, b and f. If is incident to exactly three -faces but it is not a -face, then it sends to each adjacent -face, and by a and b. If is incident to at most two -faces, then it sends at most to each adjacent -face, and by a and b.

By the same argument as in Theorem 1.2, every -face averagely sends at most via each incident edge, thus each -face has final charge . In particular, each -face has positive final charge.

The final charge of each vertex is nonnegative. Each -vertex is not involved in the discharging procedure, so its final charge is zero. Let be a -vertex incident with a face . If is neither a -face nor a -face, then sends nothing to . If is a -face, then possibly sends to by b, or sends via to a sink by c. If is a -face other than -face incident to exactly one -vertex, then sends at most to . If is a -face incident to exactly one -vertex, then sends to , but it sends nothing to/via the two -faces adjacent to , otherwise there is a -cap-subgraph. Hence, averagely sends at most to/via incident face in any case, thus .

Let be a -vertex incident to five consecutive faces and . We divide the discussions into four cases.

(i) Suppose that is incident to two adjacent -faces, say and . Since each -cycle has no chords, neither nor is a -face. If is a -face, then it is adjacent to at most three -faces by the absence of (b) and (c), but it is not a -face, and then it receives nothing from . If is a -face, then it also receives nothing from . To sum up, sends nothing to , and symmetrically sends nothing to . By the discharging rules, if is a -face, then sends nothing to ; while is a -face, then possibly sends via to a sink. Note that sends out nothing via or by the absence of (b) and (c). Hence, .

(ii) Suppose that is incident to two nonadjacent -faces, say and . If the common edge between and is -edge, then sends at most to each of and by e and f, and then sends at most in total to and . If the common edge between and is a -edge, then sends to one face in and sends nothing to the other face, or sends at most to each of and , thus sends at most in total to and . To sum up, sends at most in total to and in any case. If sends to , then must be a -face incident to exactly one -vertex and cannot be a related source of some faces, which implies that . If sends at most to , then could be related sources of two -faces, and then .

(iii) Suppose that is incident to exactly one -face . By the discharging rules, sends at most to each of and , nothing to each of and , and possibly via to a sink, which implies that .

(iv) If is not incident to any -face, then . ∎

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

Suppose that is a counterexample to Theorem 1.8 with minimum number of vertices. It is observed that is connected and is a minimal non-strictly -degenerate pair. By Theorem 1.7 and Theorem 2.2a, must contain a -cap-subgraph . Note that

 V(F5)⊆D={v∣f(v,1)+f(v,2)+⋯+f(v,κ)≥degG(v)=4}.

Moreover, is -connected and it is neither a cycle nor a complete graph. On the other hand, there exists a vertex such that , which contradicts Theorem 2.2b. ∎

###### Remark 2.

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

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

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

3. if is a planar graph without -cycles adjacent to -cycles, then it is DP--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 -cycle and an -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 , 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 be a signed planar graph and has no subgraphs isomorphic to the configurations in Fig. 3, then is signed -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).

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