A structure of 1-planar graph and its applications to coloring problems

02/24/2019 ∙ by Xin Zhang, et al. ∙ 0

A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. In this paper, we first give a useful structural theorem for 1-planar graphs, and then apply it to the list edge and list total coloring, the (p,1)-total labelling, and the equitable edge coloring of 1-planar graphs. More precisely, we verify the well-known List Edge Coloring Conjecture and List Total Coloring Conjecture for 1-planar graph with maximum degree at least 18, prove that the (p,1)-total labelling number of every 1-planar graph G is at most Δ(G)+2p-2 provided that Δ(G)≥ 8p+2 and p≥ 2, and show that every 1-planar graph has an equitable edge coloring with k colors for any integer k≥ 18. These three results respectively generalize the main theorems of three different previously published papers.

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1 Introduction

Throughout the paper, all graphs are finite, simple and undirected. By and , we denote the set of vertices, the set of edges, the minimum degree and the maximum degree of a graph . If is a plane graph, then denotes the set of faces of . A -, - and -vertex (resp. face) is a vertex (resp. face) of degree , at least and at most , respectively. For undefined concepts we refer the reader to [2].

A proper edge (resp. total) -coloring of is a function from (resp. ) to so that if and are two adjacent edges (resp. adjacent/incident elements) in . The minimum such that has a proper edge (resp. total) -coloring is the edge (resp. total) chromatic number of , denoted by (resp. ).

An edge assignment for the graph is a function so that for any edge , is a list of possible colors that can be used on . If has a proper edge coloring such that for each edge of , then we say that is edge--colorable and is an edge--coloring of . A graph is - if, whenever we give lists of colors (where is a function from to ) to each edge of , is edge--colorable. If is edge -choosable and for each edge , then is edge -choosable. The minimum such that is edge -choosable is the list edge chromatic number or edge choosability of , denoted by . The list total chromatic number or total choosability of , denoted by , is defined similarly.

Concerning the edge choosability and the total choosability of graphs, there are two well-known conjectures.

Conjecture 1.1 (List Edge Coloring Conjecture).

for any graph .

Conjecture 1.2 (List Total Coloring Conjecture).

for any graph .

The List Edge Coloring Conjecture (LECC) was independently posed by Vizing, and by Gupta, and by Albertson and Collins, and by Bollobás and Harris (see [11] for the history of this problem). The List Total Coloring Conjecture (LTCC) was posed by Borodin, Kostochka and Woodall [5]. Until now, the above two conjectures are still widely open, and particular research on some special but nontrivial classes of graphs is carried on. For example, Borodin, Kostochka and Woodall [5]

proved in 1997 that LECC and LTCC hold for planar graphs with maximum degree at least 12. Although this is a result of two decades ago, the bound 12 for the maximum degree there is still the best known bound at this moment.

The aim of this paper is to study these conjectures for the family of 1-planar graphs. A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge, and this drawing is a 1-plane graph. Usually, the number of crossings in a 1-plane graph is assumed to be as few as possible. The notion of 1-planarity was introduced by Ringel [13] while trying to simultaneously color the vertices and faces of a plane graph such that any pair of adjacent or incident elements receive different colors. Ringel [13] proved that every 1-planar graph is -colorable, and this bound for the chromatic number was later improved to 6 (being sharp) by Borodin [4, 3]. Recently in 2017, Kobourov, Liotta and Montecchiani [12] reviewed the current literature covering various research streams about 1-planarity, such as characterization and recognition, combinatorial properties, and geometric representations.

For the edge and the total colorings of 1-planar graphs, Zhang and Wu [19] proved that the edge chromatic number of every 1-planar graph with maximum degree is equal to , and Zhang and Liu [18] conjectured that the bound for can be lowered to 8, which is best possible. Zhang, Hou and Liu [17] proved that the total chromatic number of every 1-planar graph with maximum degree is at most . In 2012, Zhang, Wu and Liu [20] proved the following theorem, which confirms LECC and LTCC for 1-planar graphs with large maximum degree.

Theorem 1.1.

[20, Zhang, Wu and Liu] If is a 1-planar graph with maximum degree , then and .

A -total -labelling of a graph , introduced by Havet and Yu [8, 7], is a function from to the color set such that if , if and are two adjacent edges in , and if the vertex is incident to the edge . The minimum such that has a -total -labelling, denoted by , is the -total labelling number of . It is easy to see that . Havet and Yu [8, 9] put forward the following conjecture.

Conjecture 1.2 (-Total Labelling Conjecture).

.

For , the above conjecture is nothing but the well-known Total Coloring Conjecture, which states that . Since is a natural lower bound for , and the -total labelling is a generalization of the total coloring, it is interesting to consider when we have . Concerning this problem, Bazzaro, Montassier and Raspaud [1] proved that if is a planar graph with and , then . The lower bound for the maximum degree in this result was recently improved to by Sun and Wu [14]. For 1-planar graphs, Zhang, Yu and Liu [21] proved the following result.

Theorem 1.3.

[21, Zhang, Yu and Liu] If is a 1-planar graph with and , then .

Let be a function from to . For each vertex , let . An edge -coloring is if for each , we have

The equitable edge chromatic number of a graph is the smallest number such that has an equitable edge -coloring. However, the notion is somehow trivial since every graph has an equitable edge -coloring. Therefore, we need another notion to characterize the equitability of an edge coloring .

The equitable edge chromatic threshold of is the smallest such that has an equitable edge -coloring for any

. For example, the equitable edge chromatic threshold of any odd cycle is exactly 3.

From the above definitions, one can easily find that a proper edge coloring of is trivially equitable. Hence we immediately conclude that . However, may be a too large upper bound for . For example, Song, Wu and Liu [15] proved for series-parallel graphs that if and only if is not a connected graph with the number of edges being odd in which each vertex has even degree. Hu et al. [10] proved that for any planar graph . For 1-planar graphs, Hu et al. [10] gave the following result.

Theorem 1.4.

[10, Hu et al.] If is a 1-planar graph, then .

In this paper, we first present in Section 2 an useful structural theorem for 1-planar graphs, which can be used to consider not only the list edge and list total coloring problems, but also some other coloring problems such as the -total labelling and the equitable edge coloring. In Section 3, we prove that LECC and LTCC hold for 1-planar graphs with maximum degree at least 18, which improves Theorem 1.1. In Section 4, we consider the -total labeling of 1-planar graph by proving if and . This improves Theorem 1.3. Actually, this result also generalizes the previously mentioned result of Bazzaro, Montassier and Raspaud on planar graphs to the same result on 1-planar graphs. In Section 5, we improve the upper bound for the equitable edge chromatic threshold of 1-planar graphs in Theorem 1.4 to 18.

2 Structural Theorem

The associated plane graph of a 1-plane graph is the plane graph that is obtained from by turning all crossings of into new vertices of degree four. These new vertices in are false vertices, and the original vertices of are true ones. A face in is false if it is incident with at least one false vertex, and true otherwise.

Lemma 2.1.

[19, Lemma 1] If is a -plane graph, then

(a) false vertices in are not adjacent;

(b) false 3-face in is not incident with 2-vertex;

(c) if a -vertex is incident with two -faces and adjacent to two false vertices in , then is incident with a -face;

(d) there exists no edge in such that , is a false vertex, and is incident with two -faces.

A bipartite subgraph of is a -alternator of with partite sets for some if for each , and for each .

A bipartite subgraph of is a -alternating subgraph of with partite sets for some if for each , and for each .

Lemma 2.2.

[16, Lemma 2.4] (resp. [10, Lemma 7]) Let be a fixed integer and let be a graph without -alternator (resp. -alternating subgraph). Let and . If , then there exists a bipartite subgraph of with partite sets such that for each and for each .

Remark: The second result (while -alternating subgraph is forbidden in ) of the above lemma comes from the first three paragraphs of the proof of Lemma 7 in [10]. Although is assumed to be at most 5 in [10], the upper bound for can actually be relaxed to without changing any word in their proof.

Following Lemma 2.2, we call the - of if and . By Lemma 2.2, we conclude that

(2.1)

and

(2.2)
Theorem 2.3.

If is a 1-planar graph with minimum degree at least 2, then contains

(a) an edge with and , or

(b) an edge with and , or

(c) a -alternator (resp. -alternating subgraph) for some .

  • Suppose, to the contrary, that is a minimal counterexample (in terms of ) to this theorem. Clearly, is connected.

    If , then choose an edge of such that . Since is a 1-planar graph, (see [6]). This implies that . Hence configuration (a) or (b) occurs in , a contradiction.

    Hence, . By (2.1) and the absence of the configuration (c), each -vertex with (if it exists) of has a -master for each .

    We apply the discharging method to the associated plane graph of . Formally, for each vertex , let be its initial charge, and for each face , let be its initial charge. Clearly, by the well-known Euler’s formula.

    In what follows, we call a true vertex of big if , and small if . Since and are forbidden in , any two small vertices are not adjacent in . We use and to represent false vertex, big vertex and small vertex, respectively, and then use these notations to represent the structure of a face of . For example, we say that a face is an -face if it is a 4-face with vertices and lying cyclically on the boundary of such that is false, is small, is big, and is small.

    If a face is incident with a false vertex so that the two neighbors of in the subgraph induced by the edges of are big vertices, then is a hungry false vertex incident with . A face in is burdened if it is incident with at least one small vertex.

    We define discharging rules as follows.

    R1

    every big vertex of sends to each of its incident faces.

    R2

    every -face of sends to each of its incident hungry false vertices, and to each of its incident false vertices that are not hungry.

    R3

    every false 3-face of sends all of its received charge after applying R1 to its incident false vertex.

    R4

    every true 3-face of sends all of its received charge after applying R1 to its incident small vertex (if it exists).

    R5

    every -face of redistributes it remaining charge after applying R1 and R2 equitably to each of its incident small vertices (if it exists).

    R6

    every 2-vertex of receives and from its 2-master, 3-master, 4-master and 5-master, respectively.

    R7

    every 3-vertex of receives and from its 3-master, 4-master and 5-master, respectively.

    R8

    every 4-vertex of receives and from its 4-master and 5-master, respectively.

    R9

    every 5-vertex of receives from its 5-master.

    Here one shall note that if and , then may simultaneously be a -master of for several values with .

    Let be the charge of after applying the above rules. Since our rules only move charge around, and do not affect the sum, we have

    Next, we prove that for each . This leads to , a contradiction.

    Since every -face of is incident with at most false vertices by Lemma 2.1(a), the charge of after applying R2 is at least for . On the other hand, if is a 4-face incident with at least one hungry false vertex, then it is incident with at least two big vertices and thus by R1 and R2, and if is a 4-face incident with none hungry false vertex, then by R2. Hence, R1–R5 guarantee that for each .

    By R1, R3 and R4, it is easy to conclude the following three claims.

    Claim 1.

    Every -face sends to its incident false vertex.

    Claim 2.

    Every -face sends to its incident false vertex.

    Claim 3.

    Every burdened true 3-face sends to its incident small vertex.

    Now we consider burdened -faces.

    Claim 4.

    Every burdened 4-face sends to each of its incident small vertices if is an -face, if is an -face, if is an -face, and at least otherwise.

    • If is an -face, then the false vertices incident with are not hungry, and thus by R2 and R5, sends to each of its incident small vertices.

      If is an -face, then the false vertex incident with is not hungry, and thus by R1, R2 and R5, sends to each of its incident small vertices.

      If is an -face, then the false vertices incident with are not hungry, and thus by R1, R2 and R5, sends to its incident small vertex.

      By symmetry, can be of another types among and . In each case we can similarly calculate that sends at least to each of its incident small vertices. ∎

    Claim 5.

    Every burdened -face sends at least to each of its incident small vertices.

    • If is not incident with hungry false vertex, then is incident with at most false vertices and at most small vertices. Hence sends at least to each of its incident small vertices by R2 and R5.

      If is incident with a hungry false vertex, then is incident with at most hungry false vertices (otherwise is not burdened) and at most small vertices. By R1, R2 and R5, sends at least to each of its incident small vertices. ∎

    Now we calculate the final charge of each vertex .

    Case 1. is a false vertex.

    If is incident with at least three -faces, then by Claim 1, .

    If is incident with exactly two -faces, then each of another two faces that are incident with is an -face or a -face. Hence by Claims 1, 2 and R2, we have .

    If is incident with exactly one -face, then is incident with at least one -face, because otherwise is incident with an -face, which is impossible since small vertices are not adjacent in . Under this condition, by Claims 1, 2 and R2, we have .

    If is incident with none -face, then is incident with at least two -faces, because otherwise is incident with an -face, which is impossible since small vertices are not adjacent in . Under this condition, by Claims 1, 2 and R2, we have .

    Case 2. is a 2-vertex.

    By Lemma 2.1(b), is not incident with a false 3-face, and by R6, receives and from its 2-master, 3-master, 4-master and 5-master, respectively.

    If is incident with a true 3-face, then is adjacent to two big vertices in , and the other face incident with is either a -face, or an -face, or a -face, or a -face. In either case, sends at least to by Claims 4 and 5. Hence by Claim 3.

    If is incident with two -faces, one of which is a -face, then by Claims 4 and 5.

    If is incident with two -faces, then none of the two 4-faces incident with is an -face (otherwise a multi-edge appears in ). This implies by Claim 4.

    Case 3. is a 3-vertex.

    By R7, receives and from its 3-master, 4-master and 5-master, respectively.

    If is incident with a -face, then by Claim 5.

    If is incident with three -faces, then at most one of them is an -face (otherwise two small vertices are adjacent in ). Therefore, by Claim 4.

    If is incident with two 4-faces and one 3-face, then the two 4-faces incident with cannot be both of -type. If none of them is of -type, then by Claim 4. If one of them is of type , then the other one is of type . This implies by Claim 4.

    If is incident with one -face and two 3-faces, then the -face incident with is not of -type (otherwise a multi-edge occurs in ). If is incident with a true 3-face, then by Claims 3 and 4. If is incident with two false 3-faces, then by Lemmas 2.1(c) and 2.1(d), is adjacent to two false vertices and incident with a -face, which is impossible in this case.

    If is incident with three 3-faces, then by Lemma 2.1(d), all of those 3-faces are true. This implies by Claim 3.

    Case 4. is a true 4-vertex.

    By R8, receives and from its 4-master and 5-master, respectively.

    If is incident with at least one -face, then by Claim 5. Therefore we assume that is incident only with -faces.

    If is incident with four 3-faces, then at least two of them are true ones (otherwise two false vertices are adjacent in or there exists a multi-edge in ). Hence by Claim 3.

    If is incident with at least three -faces, then by Claim 4.

    If is incident with at exactly two -faces, then at least one of them is not of -type, which implies by Claim 4.

    If is incident with exactly one -face and this 4-face is not of -type, then by Claim 4.

    If is incident with one -face and three 3-faces, then is incident with a true 3-face. This implies that by Claims 3 and 4.

    Case 5. is a 5-vertex.

    By R9, receives from its 5-master.

    If is incident with at least one -face, then by Claim 4.

    If is incident with five 3-faces, then at least one of them is true, which implies by Claim 3.

    Case 6. is a vertex of degree between 6 and 14.

    By the absence of the configuration (a), every -vertex is adjacent only to -vertex in . Therefore, cannot be a master of any vertex. If is a small vertex, then does not give out any charge by R1–R9, and thus . If is a big vertex, that is, , then by R1, .

    Case 7. is a 15-vertex.

    By the absence of the configuration (a), is adjacent only to -vertex in . Therefore, by (2.2), can be a 5-master of at most four vertices , and cannot be a 4-master, or a 3-master, or a 2-master of any vertex. By R1 and R9, .

    Case 8. is a 16-vertex.

    By the absence of the configuration (a), is adjacent only to -vertex in . Therefore, by (2.2), can be a 5-master of at most four vertices, a 4-master of at most three vertices, and cannot be a 3-master or a 2-master of any vertex. By R1, R8 and R9, .

    Case 9. is a 17-vertex.

    By the absence of the configuration (a), is adjacent only to -vertex in . Therefore, by (2.2), can be a 5-master of at most four vertices, a 4-master of at most three vertices, a 3-master of at most two vertices, and cannot be a 2-master of any vertex. By R1, R7, R8 and R9, .

    Case 10. is a -vertex.

    By (2.2), can be a 5-master of at most four vertices, a 4-master of at most three vertices, a 3-master of at most two vertices, and a 2-master of at most one vertex. By R1, R6, R7, R8 and R9, . ∎

3 List edge and list total coloring

A critical edge -choosable graph (resp. critical total -choosable graph) is a graph with maximum degree at most such that is not edge -choosable (resp. total -choosable), and any proper subgraph of is edge -choosable (resp. total -choosable). The structures of such critical graphs were investigated by Wu and Wang [16], who proved the following two useful results.

Lemma 3.1.

[16, Lemma 2.2] If is a critical edge -choosable graph (resp. critical total -choosable graph), then for every edge with , we have .

Lemma 3.2.

[16, Lemma 2.3] If is a critical edge -choosable graph (resp. critical total -choosable graph), then there is no -alternator in for any integer .

Now we apply the above two lemmas along with Theorem 2.3 to proving the following theorem.

Theorem 3.3.

If is a 1-planar graph with maximum degree , then and .

  • Let be an integer such that and . It is sufficient to prove that and .

    Suppose, to the contrary, that there is a critical edge -choosable graph (resp. critical total -choosable graph) . By Lemma 3.1, . Since is a 1-planar graph, by Theorem 2.3, contains either (i) an edge with and , or (ii) a -alternator for some . However, Lemma 3.1 implies that the local configuration (i) is forbidden, and Lemma 3.2 implies that the local configuration (ii) is absent. This contradiction completes the proof. ∎

4 -total labelling

A critical -total -labelled graph is a graph such that it admits no -total -labelling, and any proper subgraph of has a -total -labelling. Zhang, Yu and Liu [21] proved the following two structural theorems for the critical -total labelled graph.

Lemma 4.1.

[21, Lemmas 2.1 and 2.2] Let be a critical -total -labelled graph with maximum degree at most . For any edge , if , then , and otherwise, .

Lemma 4.2.

[21, Lemma 2.4] If is a critical -total -labelled graph with maximum degree at most , then there is no -alternator in for any integer .

Theorem 4.3.

If is a 1-planar graph with and , then .

  • Let be an integer such that and . Now, proving is sufficient. Suppose, to the contrary, that is a critical -total -labelled graph. By Lemma 4.1, . Since is a 1-planar graph, by Theorem 2.3, contains either (i) an edge with and , or (b) an edge with , or (c) a -alternator for some . However, the configuration (i) or (ii) cannot appear in by Lemma 4.1, and the configuration (iii) is absent from by Lemma 4.2. ∎

5 Equitable edge coloring

A critical equitable edge -colorable graph is a graph such that admits no equitable edge -colorings, and any proper subgraph of is equitable edge -colorable. The following are two useful structural results for the critical equitable edge -colorable graph.

Lemma 5.1.

[10, Lemma 6] If is a critical equitable edge -colorable graph, then for any .

Lemma 5.2.

[10, Lemma 7] If is a critical equitable edge -colorable graph, then there is no -alternating subgraph in for any integer .

Remark: the original statements of Lemmas 6 and 7 in [10] are not as the same as the above two ones. Actually, Lemma 6 of the paper [10] states that if is a critical equitable edge -colorable graph with , then for any . Indeed, the proof there is still applicable for proving Lemma 5.1 here, only with few changes. On the other hand, from the fourth paragraph to the end of the proof of Lemma 7 in [10], the authors claim that any critical equitable edge -colorable graph does not contains a bipartite subgraph with partite sets such that for each , and for each , where . One can easily check that their proof can be directly extended to the case when , without changing any word. Therefore, there is no -alternating subgraph in a critical equitable edge -colorable graph for any integer .

Theorem 5.3.

If is a 1-planar graph, then .

  • Let be an integer such that . We just need to prove that has an equitable edge -coloring. Suppose, to the contrary, that is a critical equitable edge -colorable graph. By Lemma 5.1, . Since is a 1-planar graph, by Theorem 2.3, contains either (i) an edge with , or (ii) a -alternating subgraph for some . However, for any by Lemma 5.1, which makes the configuration (i) absent, and Lemma 5.2 do not support the appearance of the configuration (ii). ∎

References

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