# Planar graphs with $Delta geq 7$ and no triangle adjacent to a $C 4$ are minimally edge and total choosable

Planar graphs with $Delta geq 7$ and no triangle adjacent to a $C 4$ are minimally edge and total choosable - Download this document for free, or read online. Document in PDF available to download.

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique 2 G-SCOP OC - OC G-SCOP - Laboratoire des sciences pour la conception, l-optimisation et la production 3 UM3 - Université Paul-Valéry - Montpellier 3 4 ALGCO - Algorithmes, Graphes et Combinatoire LIRMM - Laboratoire d-Informatique de Robotique et de Microélectronique de Montpellier

Abstract : For planar graphs, we consider the problems of list edge coloring and list total coloring. 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.

Keywords : planar graphs Edge coloring Discharging

Author: Marthe Bonamy - Benjamin Lévêque - Alexandre Pinlou -

Source: https://hal.archives-ouvertes.fr/