On the chromatic number ofP5-free graphs with no large intersecting cliques

A graph G is called (H1, H2)-free if G contains no induced subgraph isomorphic to H(1 )or H-2. Let P(k )be a path with k vertices and C-s,C- t,C- k (s <= t) be a graph consisting of two intersecting complete graphs K(s+k )and K(t+k )with exactly k common vertices. In this paper, using an iterative method, we prove that the class of (P5,C-s,C-t,C-k)-free graphs with clique number omega has a polynomial chi-binding function f(omega) = c(s, t, k) omega(max{s, k}). In particular, we give two improved chromatic bounds: every (P5, butterfly)-free graph G has chi (G) <= 3/2 omega (G)(omega(G)-1); every (P-5,C-1,C-3)-free graph G has chi (G) <= 9 omega(G).

Automated summary

This paper studies a special class of loop-free graphs and proves that they have a polynomial chi-binding function using an iterative method. It also provides two improved chromatic bounds.