Abstract
For an arbitrary class of graphs \(\mathcal{G}\), there may not exist a function f such that \(\chi(G) \leq f(\omega(G))\), for every \(G \in \mathcal{G}\). When such a function exists, it is called a χ-binding function for \(\mathcal{G}\). The problem of finding an optimal χ-binding function for the class of 3K 1-free graphs is open. In this paper, we obtain linear χ-binding function for the class of {3K 1, H}-free graphs, where H is one of the following graphs: \(K_1 + C_4, (K_3 \cup K_1)+K_1, K_1 + P_4, K_4 \cup K_1\), House graph and Kite graph. We first describe structures of these graphs and then derive χ-binding functions.
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Choudum, S.A., Karthick, T. & Shalu, M.A. Linear Chromatic Bounds for a Subfamily of 3K 1-free Graphs. Graphs and Combinatorics 24, 413–428 (2008). https://doi.org/10.1007/s00373-008-0801-0
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DOI: https://doi.org/10.1007/s00373-008-0801-0