A hole is a chordless cycle with at least four vertices. A hole is even if its number of vertices is even. Given a set L of graphs, a graph G is L-free if G does not contain any graph in L as an induced subgraph. Currently, the following two problems are unresolved: the complexity of coloring even hole-free graphs, and the complexity of coloring \((4K_1, C_4)\)-free graphs. The intersection of these two problems is the problem of coloring \((4K_1, C_4, C_6)\)-free graphs. In this paper we present partial results on this problem.
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We note that the definition of near-uniform partition in  is incomplete. The sets \(S_i\)’s must be cliques for the theorem to hold.
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This work was supported by the Canadian Tri-Council Research Support Fund. The authors A.M.F. and C.T.H. were each supported by individual NSERC Discovery Grants. Author T.P.M was supported by an NSERC Undergraduate Student Research Award (USRA). This work was done by authors D.J.F. and K.H. in partial fulfillment of the course requirements for CP493: Directed Research Project I in the Department of Physics and Computer Science at Wilfrid Laurier University.
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Angèle M. Foley: formerly Angèle M. Hamel.
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Foley, A.M., Fraser, D.J., Hoàng, C.T. et al. The Intersection of Two Vertex Coloring Problems. Graphs and Combinatorics 36, 125–138 (2020). https://doi.org/10.1007/s00373-019-02123-1
- Graph coloring
- Perfect graphs