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 edgecolorable and is an edgecoloring of . A graph is  if, whenever we give lists of colors (where is a function from to ) to each edge of , is edgecolorable. 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 wellknown 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 1planar graphs. A graph is 1planar 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 1plane graph. Usually, the number of crossings in a 1plane graph is assumed to be as few as possible. The notion of 1planarity 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 1planar 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 1planarity, such as characterization and recognition, combinatorial properties, and geometric representations.
For the edge and the total colorings of 1planar graphs, Zhang and Wu [19] proved that the edge chromatic number of every 1planar 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 1planar graph with maximum degree is at most . In 2012, Zhang, Wu and Liu [20] proved the following theorem, which confirms LECC and LTCC for 1planar graphs with large maximum degree.
Theorem 1.1.
[20, Zhang, Wu and Liu] If is a 1planar 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 wellknown 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 1planar graphs, Zhang, Yu and Liu [21] proved the following result.
Theorem 1.3.
[21, Zhang, Yu and Liu] If is a 1planar 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 seriesparallel 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 1planar graphs, Hu et al. [10] gave the following result.
Theorem 1.4.
[10, Hu et al.] If is a 1planar graph, then .
In this paper, we first present in Section 2 an useful structural theorem for 1planar 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 1planar graphs with maximum degree at least 18, which improves Theorem 1.1. In Section 4, we consider the total labeling of 1planar 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 1planar graphs. In Section 5, we improve the upper bound for the equitable edge chromatic threshold of 1planar graphs in Theorem 1.4 to 18.
2 Structural Theorem
The associated plane graph of a 1plane 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 3face in is not incident with 2vertex;
(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.
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.
Theorem 2.3.
If is a 1planar 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 1planar 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 wellknown 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 4face 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 3face of sends all of its received charge after applying R1 to its incident false vertex.
 R4

every true 3face 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 2vertex of receives and from its 2master, 3master, 4master and 5master, respectively.
 R7

every 3vertex of receives and from its 3master, 4master and 5master, respectively.
 R8

every 4vertex of receives and from its 4master and 5master, respectively.
 R9

every 5vertex of receives from its 5master.
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 4face 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 4face 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 3face sends to its incident small vertex.
Now we consider burdened faces.
Claim 4.
Every burdened 4face 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 2vertex.
By Lemma 2.1(b), is not incident with a false 3face, and by R6, receives and from its 2master, 3master, 4master and 5master, respectively.
If is incident with a true 3face, 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, then none of the two 4faces incident with is an face (otherwise a multiedge appears in ). This implies by Claim 4.
Case 3. is a 3vertex.
By R7, receives and from its 3master, 4master and 5master, 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 4faces and one 3face, then the two 4faces 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 3faces, then the face incident with is not of type (otherwise a multiedge occurs in ). If is incident with a true 3face, then by Claims 3 and 4. If is incident with two false 3faces, 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 3faces, then by Lemma 2.1(d), all of those 3faces are true. This implies by Claim 3.
Case 4. is a true 4vertex.
By R8, receives and from its 4master and 5master, 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 3faces, then at least two of them are true ones (otherwise two false vertices are adjacent in or there exists a multiedge 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 4face is not of type, then by Claim 4.
If is incident with one face and three 3faces, then is incident with a true 3face. This implies that by Claims 3 and 4.
Case 5. is a 5vertex.
By R9, receives from its 5master.
If is incident with at least one face, then by Claim 4.
If is incident with five 3faces, 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 15vertex.
By the absence of the configuration (a), is adjacent only to vertex in . Therefore, by (2.2), can be a 5master of at most four vertices , and cannot be a 4master, or a 3master, or a 2master of any vertex. By R1 and R9, .
Case 8. is a 16vertex.
By the absence of the configuration (a), is adjacent only to vertex in . Therefore, by (2.2), can be a 5master of at most four vertices, a 4master of at most three vertices, and cannot be a 3master or a 2master of any vertex. By R1, R8 and R9, .
Case 9. is a 17vertex.
By the absence of the configuration (a), is adjacent only to vertex in . Therefore, by (2.2), can be a 5master of at most four vertices, a 4master of at most three vertices, a 3master of at most two vertices, and cannot be a 2master of any vertex. By R1, R7, R8 and R9, .
Case 10. is a vertex.
By (2.2), can be a 5master of at most four vertices, a 4master of at most three vertices, a 3master of at most two vertices, and a 2master 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 1planar 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 1planar 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 1planar 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 1planar 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 1planar 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 1planar 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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