"Coloring Graph Classes with no Induced Fork via Perfect Divisibility" by T. Karthick, Jenny Kaufmann et al.
 

Coloring Graph Classes with no Induced Fork via Perfect Divisibility

Article Type

Research Article

Publication Title

Electronic Journal of Combinatorics

Abstract

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 K1,3 by subdividing an edge once. The problem of finding a quadratic 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.

DOI

10.37236/10348

Publication Date

1-1-2022

Comments

Open Access, Gold, Green

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