Adjacent vertex distinguishing total coloring of planar graphs with large maximum degree

A total k-coloring of a graph is an assignment of k colors to its vertices and edges such that no two adjacent or incident elements receive the same color. The total coloring conjecture (TCC) states that every simple graph G has a total ΔG+2-coloring, where ΔG is the maximum degree of G. This conjecture has been confirmed for planar graphs with maximum degree at least 7 or at most 5, i.e., the only open case of TCC is that of maximum degree 6. It is known that every planar graph G of ΔG≥9 or ΔG∈7,8 with some restrictions has a total ΔG+1-coloring. In particular, in (Shen and Wang, 2009), the authors proved that every planar graph with maximum degree 6 and without 4-cycles has a total 7-coloring. In this paper, we improve this result by showing that every diamond-free and house-free planar graph of maximum degree 6 is totally 7-colorable if every 6-vertex is not incident with two adjacent four cycles or three cycles of size p,q,ℓ for some p,q,ℓ∈3,4,4,3,3,4.

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Total coloring of planar graphs with maximum degree 8

Theoretical Computer Science ◽

10.1016/j.tcs.2013.12.006 ◽

2014 ◽

Vol 522 ◽

pp. 54-61 ◽

Cited By ~ 4

Author(s):

Huijuan Wang ◽

Lidong Wu ◽

Jianliang Wu

Keyword(s):

Planar Graphs ◽

Maximum Degree ◽

Total Coloring

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The adjacent vertex distinguishing total coloring of planar graphs

Journal of Combinatorial Optimization ◽

10.1007/s10878-012-9527-2 ◽

2012 ◽

Vol 27 (2) ◽

pp. 379-396 ◽

Cited By ~ 34

Author(s):

Weifan Wang ◽

Danjun Huang

Keyword(s):

Planar Graphs ◽

Total Coloring ◽

Adjacent Vertex

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Planar graphs with $\Delta \geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2147 ◽

2016 ◽

Vol Vol. 17 no. 3 (Graph Theory) ◽

Author(s):

Marthe Bonamy ◽

Benjamin Lévêque ◽

Alexandre Pinlou

Keyword(s):

Planar Graphs ◽

Maximum Degree ◽

Edge Coloring ◽

Total Coloring ◽

List Total Coloring ◽

List Edge Coloring ◽

International Audience

International audience For planar graphs, we consider the problems of <i>list edge coloring</i> and <i>list total coloring</i>. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors. Total coloring is the problem of coloring the edges and the vertices while ensuring that two edges that are adjacent, two vertices that are adjacent, or a vertex and an edge that are incident receive different colors. In their list extensions, instead of having the same set of colors for the whole graph, every vertex or edge is assigned some set of colors and has to be colored from it. A graph is minimally edge or total choosable if it is list $\Delta$-edge-colorable or list $(\Delta +1)$-total-colorable, respectively, where $\Delta$ is the maximum degree in the graph. It is already known that planar graphs with $\Delta \geq 8$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable (Li Xu 2011), and that planar graphs with $\Delta \geq 7$ and no triangle sharing a vertex with a $C_4$ or no triangle adjacent to a $C_k (\forall 3 \leq k \leq 6)$ are minimally total colorable (Wang Wu 2011). We strengthen here these results and prove that planar graphs with $\Delta \geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable.

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Total coloring of planar graphs of maximum degree eight

Information Processing Letters ◽

10.1016/j.ipl.2010.02.012 ◽

2010 ◽

Vol 110 (8-9) ◽

pp. 321-324 ◽

Cited By ~ 12

Author(s):

Nicolas Roussel ◽

Xuding Zhu

Keyword(s):

Planar Graphs ◽

Maximum Degree ◽

Total Coloring

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