Elsevier

Discrete Mathematics

Volume 207, Issues 1–3, 28 September 1999, Pages 257-262
Discrete Mathematics

Near-complete multipartite graphs and forbidden induced subgraphs

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Abstract

A proper vertex k-coloring C1,C2,…,Ck of a graph G is called l-bounded (l⩾0) if |CiN(u)|⩽l for each i=1,2,…,k and each vertex uVGCi, where N(u) is the neighborhood of u. Let C(k,l) be the class of all graphs having an l-bounded k-coloring (k⩾1 and l⩾0).

We prove that every class C(k,l) has a finite forbidden induced subgraph characterization. This result implies the existence of polynomial algorithms for recognition of C(k,l). The set of all 14 minimal forbidden induced subgraphs for C(3,1) is found.

Keywords

Vertex coloring
Hereditary classes
Forbidden induced subgraphs

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