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On S-packing edge-colorings of graphs with small edge weight

Author

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  • Yang, Wei
  • Wu, Baoyindureng

Abstract

The edge weight, denoted by w(e), of a graph G is max{dG(u)+dG(v):uv∈E(G)}. For an integer sequence S=(s1,s2,…,sk) with 0≤s1≤s2≤⋯≤sk, an S-packing edge-coloring of a graph G is a partition of E(G) into k subsets E1,E2,…,Ek such that for each 1≤i≤k, dL(G)(e,e′)≥si+1 for any e,e′∈Ei, where dL(G)(e,e′) denotes the distance of e and e′ in the line graph L(G) of G. Hocquard, Lajou and Lužar (Between proper and strong edge-colorings of subcubic graphs, https://arxiv.org/abs/2011.02175) posed an open problem: every subcubic bipartite graph G with w(e)≤5 is (1,24)-packing edge-colorable. We confirm the question in affirmative with a stronger way. It is shown that for any graph G (not necessarily subcubic bipartite) with w(e)≤5 is (1,24)-packing edge-colorable. We also prove that every graph G with w(e)≤6 is (1,28)-packing edge-colorable.

Suggested Citation

  • Yang, Wei & Wu, Baoyindureng, 2022. "On S-packing edge-colorings of graphs with small edge weight," Applied Mathematics and Computation, Elsevier, vol. 418(C).
    • Handle: RePEc:eee:apmaco:v:418:y:2022:i:c:s0096300321009231
    DOI: 10.1016/j.amc.2021.126840
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