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On the Structure of Graphs Without Claw, \(4K_1\) and co-R

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Abstract

Given a family F of graphs, a graph G is F-free if it does not contain any graph in F as an induced subgraph. The problem of determining the complexity of vertex coloring (claw, \(4K_1\))-free graphs is a well known open problem. In this paper, we solve the coloring problem for a subclass of (claw, \(4K_1\))-free graphs. We design a polynomial-time algorithm to color (\(claw\), \(4K_1\), \(co\)-\(R\))-free graphs. This algorithm is derived from a structural theorem on (\(claw\), \(4K_1\), \(co\)-\(R\))-free graphs.

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Notes

  1. Find a maximum matching in the complement of G, say this matching has m edges, then we know that \(\chi (G) = |V(G) | - m\)

  2. Actually it can be proved that \(cwd(P_5) = 3\).

  3. If \(C_n\) has exactly two vertices, then the labelling will be obvious from the algorithm’s description

  4. For the labelled graph (G, L), the labels need not be integers

  5. The proof of Lemma 3.37 also follows from application of Theorem 2.7 on the sets \(X_1, \dots , X_5\)

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Acknowledgements

This work was supported by the Canadian Tri-Council Research Support Fund. The author C.T.H. was supported by an individual NSERC Discovery Grant. Author T.A. was supported by an NSERC Undergraduate Student Research Award (USRA).

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This work is supported by the Canadian Tri-Council Research Support Fund.

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

    Tala Abuadas & Chính T. Hoàng

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  1. Tala Abuadas
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Abuadas, T., Hoàng, C.T. On the Structure of Graphs Without Claw, \(4K_1\) and co-R. Graphs and Combinatorics 38, 123 (2022). https://doi.org/10.1007/s00373-022-02517-8

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  • DOI: https://doi.org/10.1007/s00373-022-02517-8

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