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On the Structure of (Even Hole, Kite)-Free Graphs

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Abstract

A hole is an induced cycle with at least four vertices. A hole is even if it has an even number of vertices. Even-hole-free graphs are being actively studied in the literature. It is not known whether even-hole-free graphs can be optimally colored in polynomial time. A diamond is the graph obtained from the clique of four vertices by removing one edge. A kite is a graph consists of a diamond and another vertex adjacent to a vertex of degree two in the diamond. Kloks et al. (J Combin Theory B 99:733–800, 2009) designed a polynomial-time algorithm to color an (even hole, diamond)-free graph. In this paper, we consider the class of (even hole, kite)-free graphs which generalizes (even hole, diamond)-free graphs. We prove that a connected (even hole, kite)-free graph is diamond-free, or the join of a clique and a diamond-free graph, or contains a clique cutset. This result, together with the result of Kloks et al., imply the existence of a polynomial time algorithm to color (even hole, kite)-free graphs. We also prove (even hole, kite)-free graphs are \(\beta \)-perfect in the sense of Markossian, Gasparian and Reed. Finally, we establish the Vizing bound for (even hole, kite)-free graphs.

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Acknowledgements

This work was supported by the Canadian Tri-Council Research Support Fund. The authors A.M.H. and C.T.H. were each supported by individual NSERC Discovery Grants. Author D.J.F. was supported by an NSERC Undergraduate Student Research Award (USRA).

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  1. Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, ON, Canada

    Dallas J. Fraser, Angèle M. Hamel & Chính T. Hoàng

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  1. Dallas J. Fraser
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  2. Angèle M. Hamel
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  3. Chính T. Hoàng
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Correspondence to Chính T. Hoàng.

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Fraser, D.J., Hamel, A.M. & Hoàng, C.T. On the Structure of (Even Hole, Kite)-Free Graphs. Graphs and Combinatorics 34, 989–999 (2018). https://doi.org/10.1007/s00373-018-1925-5

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  • DOI: https://doi.org/10.1007/s00373-018-1925-5

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