On (s, t)-relaxed strong edge-coloring of graphs

Dan He () and Wensong Lin ()
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Dan He: Southeast University
Wensong Lin: Southeast University

Journal of Combinatorial Optimization, 2017, vol. 33, issue 2, No 15, 609-625

Abstract: Abstract In this paper, we introduce a new relaxation of strong edge-coloring. Let G be a graph. For two nonnegative integers s and t, an (s, t)-relaxed strong k-edge-coloring is an assignment of k colors to the edges of G, such that for any edge e, there are at most s edges adjacent to e and t edges which are distance two apart from e assigned the same color as e. The (s, t)-relaxed strong chromatic index, denoted by $${\chi '}_{(s,t)}(G)$$ χ ′ ( s , t ) ( G ) , is the minimum number k of an (s, t)-relaxed strong k-edge-coloring admitted by G. This paper studies the (s, t)-relaxed strong edge-coloring of graphs, especially trees. For a tree T, the tight upper bounds for $${\chi '}_{(s,0)}(T)$$ χ ′ ( s , 0 ) ( T ) and $${\chi '}_{(0,t)}(T)$$ χ ′ ( 0 , t ) ( T ) are given. And the (1, 1)-relaxed strong chromatic index of an infinite regular tree is determined. Further results on $${\chi '}_{(1,0)}(T)$$ χ ′ ( 1 , 0 ) ( T ) are also presented.

Keywords: Strong edge-coloring; Strong chromatic index; $$(s; t)$$ ( s; t ) -relaxed strong edge-coloring; $$(s; t)$$ ( s; t ) -relaxed strong chromatic index; Tree; Infinite $$\Delta $$ Δ -regular tree; 05C15 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10878-015-9983-6

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