Total-chromatic number and chromatic index of dually chordal graphs

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Abstract

Given a graph G and a vertex v, a vertex u∈N(v) is a maximum neighbor of v if for all w∈N(v) we have N(w)⫅N(u), where N(v) denotes the neighborhood of v in G. A maximum neighborhood elimination order of G is a linear order v1,v2,…,vn on its vertex set such that there is a maximum neighbor of vi in the subgraph G[v1,…,vi]. A graph is dually chordal if it admits a maximum neighborhood elimination order. Alternatively, a graph is dually chordal if it is the clique graph of a chordal graph. The class of dually chordal graphs generalizes known subclasses of chordal graphs such as doubly chordal graphs, strongly chordal graphs, interval graphs, and indifference graphs. We prove that Vizing's total-color conjecture holds for dually chordal graphs. We describe a new heuristic that yields an exact total coloring for even maximum degree dually chordal graphs and an exact edge coloring for odd maximum degree dually chordal graphs.

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This work was partially supported by CNPq, PRONEX/FINEP, FAPESP and FAPERJ, Brazilian research agencies.

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