A finite basis characterization of α-split colorings

Authors:
Andras Gyarfas;Andre E. Kezdy;Jeno Lehel
Affiliations:
Computer and Automation Institute, Hungarian Academy of Sciences, 1111 Budapest, Hungary;Department of Mathematics, University of Louisville, Louisville, KY;Department of Mathematics, University of Louisville, Louisville, KY
Venue:
Discrete Mathematics - Kleitman and combinatorics: a celebration
Year:
2002

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Abstract

Fix t 1, a positive integer, and a= (a1,..., at) a vector of nonnegative integers. A t-coloring of the edges of a complete graph is called a-split if there exists a partition of the vertices into t sets V1,..., Vt such that every set of ai + 1 vertices in Vi contains an edge of color i, for i =1,...,t. We combine a theorem of Deza with Ramsey's theorem to prove that, for any fixed a, the family of a-split colorings is characterized by a finite list of forbidden induced subcolorings. A similar hypergraph version follows from our proofs. These results generalize previous work by Kézdy et al. (J. Combin. Theory Ser. A 73(2) (1996) 353) and Gyárfás (J. Combin. Theory Ser. A 81(2) (1998) 255). We also consider other notions of splitting.