Perfect divisibility and coloring of some fork-free graphs
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A hole is an induced cycle of length at least four, and an odd hole is a hole of odd length. A fork is a graph obtained from K_1,3 by subdividing an edge once. An odd balloon is a graph obtained from an odd hole by identifying respectively two consecutive vertices with two leaves of K_1, 3. A gem is a graph that consists of a P_4 plus a vertex adjacent to all vertices of the P_4. A butterfly is a graph obtained from two traingles by sharing exactly one vertex. A graph G is perfectly divisible if for each induced subgraph H of G, V(H) can be partitioned into A and B such that H[A] is perfect and ω(H[B])<ω(H). In this paper, we show that (odd balloon, fork)-free graphs are perfectly divisible (this generalizes some results of Karthick et al). As an application, we show that χ(G)≤ω(G)+12 if G is (fork, gem)-free or (fork, butterfly)-free.
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