Coloring graph classes with no induced fork via perfect divisibility
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For a graph G, χ(G) will denote its chromatic number, and ω(G) its clique number. A graph G is said to be perfectly divisible if for all induced subgraphs H of G, V(H) can be partitioned into two sets A, B such that H[A] is perfect and ω(H[B]) < ω(H). An integer-valued function f is called a χ-binding function for a hereditary class of graphs C if χ(G) ≤ f(ω(G)) for every graph G∈ C. The fork is the graph obtained from the complete bipartite graph K_1,3 by subdividing an edge once. The problem of finding a polynomial χ-binding function for the class of fork-free graphs is open. In this paper, we study the structure of some classes of fork-free graphs; in particular, we study the class of (fork,F)-free graphs G in the context of perfect divisibility, where F is a graph on five vertices with a stable set of size three, and show that every G∈ G satisfies χ(G)≤ω(G)^2. We also note that the class G does not admit a linear χ-binding function.
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