List strong edge coloring of planar graphs with maximum degree 4

Ming Chen; Jie Hu; Xiaowei Yu; Shan Zhou

doi:10.1016/j.disc.2018.10.034

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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On the Strong Chromatic Index of Sparse Graphs

The Electronic Journal of Combinatorics ◽

10.37236/5390 ◽

2018 ◽

Vol 25 (3) ◽

Author(s):

Philip DeOrsey ◽

Michael Ferrara ◽

Nathan Graber ◽

Stephen G. Hartke ◽

Luke L. Nelsen ◽

...

Keyword(s):

Planar Graph ◽

Planar Graphs ◽

Maximum Degree ◽

Applied Mathematics ◽

Edge Coloring ◽

Combinatorial Nullstellensatz ◽

Chromatic Index ◽

Sparse Graphs ◽

Strong Chromatic Index ◽

General Bound

The strong chromatic index of a graph $G$, denoted $\chi'_s(G)$, is the least number of colors needed to edge-color $G$ so that edges at distance at most two receive distinct colors. The strong list chromatic index, denoted $\chi'_{s,\ell}(G)$, is the least integer $k$ such that if arbitrary lists of size $k$ are assigned to each edge then $G$ can be edge-colored from those lists where edges at distance at most two receive distinct colors.We use the discharging method, the Combinatorial Nullstellensatz, and computation to show that if $G$ is a subcubic planar graph with ${\rm girth}(G) \geq 41$ then $\chi'_{s,\ell}(G) \leq 5$, answering a question of Borodin and Ivanova [Precise upper bound for the strong edge chromatic number of sparse planar graphs, Discuss. Math. Graph Theory, 33(4), (2014) 759--770]. We further show that if $G$ is a subcubic planar graph and ${\rm girth}(G) \geq 30$, then $\chi_s'(G) \leq 5$, improving a bound from the same paper.Finally, if $G$ is a planar graph with maximum degree at most four and ${\rm girth}(G) \geq 28$, then $\chi'_s(G)N \leq 7$, improving a more general bound of Wang and Zhao from [Odd graphs and its applications to the strong edge coloring, Applied Mathematics and Computation, 325 (2018), 246-251] in this case.

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Clustered 3-colouring graphs of bounded degree

Combinatorics Probability Computing ◽

10.1017/s0963548321000213 ◽

2021 ◽

pp. 1-13

Author(s):

Vida Dujmović ◽

Louis Esperet ◽

Pat Morin ◽

Bartosz Walczak ◽

David R. Wood

Keyword(s):

Planar Graphs ◽

Maximum Degree ◽

Bounded Degree ◽

Bounded Number ◽

Proper Vertex ◽

Vertex Colouring

Abstract A (not necessarily proper) vertex colouring of a graph has clustering c if every monochromatic component has at most c vertices. We prove that planar graphs with maximum degree $\Delta$ are 3-colourable with clustering $O(\Delta^2)$ . The previous best bound was $O(\Delta^{37})$ . This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $\Delta$ that exclude a fixed minor are 3-colourable with clustering $O(\Delta^5)$ . The best previous bound for this result was exponential in $\Delta$ .

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