A note on orientation and chromatic number of graphs

Manouchehr Zaker ()
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Manouchehr Zaker: Institute for Advanced Studies in Basic Sciences

Journal of Combinatorial Optimization, 2017, vol. 34, issue 2, No 21, 605-611

Abstract: Abstract Let D be any edge orientation of a graph G. We denote by $$\Delta _k(D)$$ Δ k ( D ) the maximum value t for which there exists a directed path $$v_1, \ldots , v_k$$ v 1 , … , v k such that $$d^{out}(v_k)=t$$ d o u t ( v k ) = t , where $$d^{out}(v_k)$$ d o u t ( v k ) is the out-degree of $$v_k$$ v k in D. We first obtain some bounds for the chromatic number of G in terms of $$\Delta _k(D)$$ Δ k ( D ) and then show a relationship between $$\Delta _k(D)$$ Δ k ( D ) and vertex partitions of a graph into degenerate subgraphs.

Keywords: Graph coloring; Chromatic number; Acyclic orientation; Degenerate subgraph; 05C15; 05C20 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10878-016-0094-9

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