Coloring hypercomplete and hyperpath graphs

Abstract

Given a graph G with an induced subgraph H and a family F of graphs, we introduce a (hyper)graph HH(G;F)=(VH, EH), the hyper-H (hyper)graph of G with respect to F, whose vertices are induced copies of H in G, and \{H1,H2,\ldots,Hr\} \in EH if and only if the induced subgraph of G by the set \cupi=1r Hi is isomorphic to a graph F in the family F, and the integer r is the least integer for F with this property. When H is a k-complete or a k-path of G, we abbreviate HKk(G;F) and HPk(G;F) to Hk(G;F) and HPk(G;F), respectively. Our motivation to introduce this new (hyper)graph operator on graphs comes from the fact that the graph Hk(Kn;\{K2k\}) is isomorphic to the ordinary Kneser graph K(n;k) whenever 2k \leq n. As a generalization of the Lovasz--Kneser theorem, we prove that c(Hk(G;\{K2k\}))=c(G)-2k+2 for any graph G with w(G)=c(G) and any integer k\leq \lfloor w(G)/2\rfloor. We determine the clique and fractional chromatic numbers of Hk(G;\{K2k\}), and we consider the generalized Johnson graphs Hr(H;\{Kr+1\}) and show that c(Hr(H;\{Kr+1\}))\leq c(H) for any graph H and any integer r< w(H). By way of application, we construct examples of graphs such that the gap between their chromatic and fractional chromatic numbers is arbitrarily large. We further analyze the chromatic number of hyperpath (hyper)graphs HPk(G;Pm), and we provide upper bounds when m=k+1 and m=2k in terms of the k-distance chromatic number of the source graph.