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 \({\chi '}_{(s,t)}(G)\), 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 \({\chi '}_{(s,0)}(T)\) and \({\chi '}_{(0,t)}(T)\) are given. And the (1, 1)-relaxed strong chromatic index of an infinite regular tree is determined. Further results on \({\chi '}_{(1,0)}(T)\) are also presented.
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Acknowledgments
Supported by Natural Science Foundation of Jiangsu Province of China (No. BK20151399).
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He, D., Lin, W. On (s, t)-relaxed strong edge-coloring of graphs. J Comb Optim 33, 609–625 (2017). https://doi.org/10.1007/s10878-015-9983-6
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DOI: https://doi.org/10.1007/s10878-015-9983-6
Keywords
- Strong edge-coloring
- Strong chromatic index
- \((s, t)\)-relaxed strong edge-coloring
- \((s, t)\)-relaxed strong chromatic index
- Tree
- Infinite \(\Delta \)-regular tree