Abstract
For two disjoint vertex subsets X, Y of a graph G, we denote \(X \leftarrow Y\) if every vertex of Y has at most one non-neighbour in X. A k-clique star partition of a graph G is \(V(G)=Q_1\cup Q_2\cup \ldots \cup Q_k\) such that (i) \(Q_{i}\) is a clique in G for all \(1\le i \le k\) and (ii) \(Q_i\leftarrow Q_j\) for all \(1 \le i < j\le k\). We prove that (a) every \(\{3K_1, 2K_2 \}\)-free graph admits a \(4\omega (G)\)-clique star partition and (b) if G is a graph with girth at least five, then its star chromatic number \(\chi _s (G)\) satisfies \(\chi _s (G) \le 4 \alpha (G)\).
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The authors wish to thank the anonymous referees whose suggestions improved the presentation of this paper.
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Shalu, M.A., Sandhya, T.P. Star Coloring of Graphs with Girth at Least Five. Graphs and Combinatorics 32, 2121–2134 (2016). https://doi.org/10.1007/s00373-016-1702-2
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DOI: https://doi.org/10.1007/s00373-016-1702-2