A note on generalized chromatic number and generalized girth

Erdős proved that there are graphs with arbitrarily large girth and chromatic number. We study the extension of this for generalized chromatic numbers. Generalized graph coloring describes the partitioning of the vertices into classes whose induced subgraphs satisfy particular constraints. When P is a family of graphs, the P chromatic number of a graph G, written χP, is the minimum size of a partition of V (G) into classes that induce subgraphs of G belonging to P. When P is the family of independent sets, χP is the ordinary chromatic number. General aspects are studied in [1-3,7-9,11-14,1718]. Many additional results are known about particular generalized chromatic numbers. One aim in the study of generalized chromatic numbers is the extension of classical coloring results. Erdős [4] proved that there exist graphs of large chromatic number and large girth. We study the extension of this for a class of generalized coloring parameters. We consider the family P consisting of all graphs not containing H as a subgraph; we call the corresponding parameter the H-chromatic number and write it as χH . The natural extension requires an appropriate definition for generalized girth. For j ≥ 2, an (H, j)-cycle in a graph G is a list of distinct subgraphs H1, . . . , Hj, each isomorphic to H, such that ⋃j i=1 Hi contains a cycle that decomposes into j nontrivial paths with the ith path in Hi (any two successive paths in the decomposition share one vertex). The H-girth of G, written gH(G), is the minimum j such that G contains an (H, j)-cycle, if this exists; otherwise gH(G) = ∞. One might prefer a weaker notion of cycle. For j ≥ 2, a weak (H, j)-cycle in G is a list of distinct subgraphs H1, . . . , Hj, each isomorphic to H, and a selection of distinct vertices † Research supported in part by NSA/MSP Grant MDA904-93-H-3040. Running head: CHROMATIC NUMBER AND GIRTH AMS codes: 05C15, 05C65, 05C80