Minimum total coloring of planar graph

Huijuan Wang (), Lidong Wu (), Weili Wu (), Panos Pardalos () and Jianliang Wu ()

Journal of Global Optimization, 2014, vol. 60, issue 4, 777-791

Abstract: Graph coloring is an important tool in the study of optimization, computer science, network design, e.g., file transferring in a computer network, pattern matching, computation of Hessians matrix and so on. In this paper, we consider one important coloring, vertex coloring of a total graph, which is familiar to us by the name of “total coloring”. Total coloring is a coloring of $$V\cup {E}$$ V ∪ E such that no two adjacent or incident elements receive the same color. In other words, total chromatic number of $$G$$ G is the minimum number of disjoint vertex independent sets covering a total graph of $$G$$ G . Here, let $$G$$ G be a planar graph with $$\varDelta \ge 8$$ Δ ≥ 8 . We proved that if for every vertex $$v\in V$$ v ∈ V , there exists two integers $$i_{v},j_{v} \in \{3,4,5,6,7,8\}$$ i v , j v ∈ { 3 , 4 , 5 , 6 , 7 , 8 } such that $$v$$ v is not incident with intersecting $$i_v$$ i v -cycles and $$j_v$$ j v -cycles, then the vertex chromatic number of total graph of $$G$$ G is $$\varDelta +1$$ Δ + 1 , i.e., the total chromatic number of $$G$$ G is $$\varDelta +1$$ Δ + 1 . Copyright Springer Science+Business Media New York 2014

Keywords: Planar graph; Total coloring; Cycle; Independent set (search for similar items in EconPapers)
Date: 2014
References: View complete reference list from CitEc
Citations: View citations in EconPapers (7)

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DOI: 10.1007/s10898-013-0138-y

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