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A note on orientation and chromatic number of graphs

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Abstract

Let D be any edge orientation of a graph G. We denote by \(\Delta _k(D)\) the maximum value t for which there exists a directed path \(v_1, \ldots , v_k\) such that \(d^{out}(v_k)=t\), where \(d^{out}(v_k)\) is the out-degree of \(v_k\) in D. We first obtain some bounds for the chromatic number of G in terms of \(\Delta _k(D)\) and then show a relationship between \(\Delta _k(D)\) and vertex partitions of a graph into degenerate subgraphs.

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Acknowledgements

The author thanks useful comments of anonymous reviewers of the paper.

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Correspondence to Manouchehr Zaker.

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Zaker, M. A note on orientation and chromatic number of graphs. J Comb Optim 34, 605–611 (2017). https://doi.org/10.1007/s10878-016-0094-9

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  • DOI: https://doi.org/10.1007/s10878-016-0094-9

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