Total-chromatic number and chromatic index of dually chordal graphs

Given a graph G and a vertex nu, a vertex u is an element of N(nu) is a maximum neighbor of nu if for all w is an element of N(nu) we have N(zu) subset of or equal to N(u), where N(nu) denotes the neighborhood of nu in G. A maximum neighborhood elimination order of G is a linear order nu(1), nu(2),..., nu(n) on its vertex set such that there is a maximum neighbor of v(i) in the subgraph G[v(1),..., v(i)]. 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 Fields an exact total coloring for even maximum degree dually chordal graphs and an exact edge coloring for odd maximum degree dually chordal graphs. (C) 1999 Elsevier Science B.V. All rights reserved.70314715

Even and odd pairs in comparability and in P-4-comparability graphs

We characterize even and odd pairs in comparability and in P-4-comparability graphs. The characterizations lead to simple algorithms for deciding whether a given pair of vertices forms an even or odd pair in these classes of graphs. The complexities of the proposed algorithms are O(n+m) for comparability graphs and O(n(2)m) for P-4-comparability graphs. The former represents an improvement over a recent algorithm of complexity O(nm). (C) 1999 Elsevier Science B.V. All rights reserved.914169929329

The non planar vertex deletion of C-n x C-m

The non planar vertex deletion or vertex deletion vd(G) of a graph G = (V, E) is the smallest non negative integer k, such that the removal of k vertices from G produces a planar graph. Hence, the maximum planar induced subgraph of G has precisely vertical bar V vertical bar - vd(G) vertices. The problem of computing vertex deletion is in general very hard, it is NP-complete. In this paper we compute the non planar vertex deletion for the family of toroidal graphs C-n x C-m.7632