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On decomposing multigraphs into locally irregular submultigraphs
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DOI：10.1016/j.amc.2023.128049.png)
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摘要
En 中文
A locally irregular multigraph is a multigraph whose adjacent vertices have distinct degrees. The locally irregular edge coloring is an edge coloring of a multigraph G such that every color induces a locally irregular submultigraph of G . We say that a multigraph G is locally irregular colorable if it admits a locally irregular edge coloring and we denote by lir (G ) the locally irregular chromatic index of G , which is the smallest number of colors required in a locally irregular edge coloring of a locally irregular colorable multigraph G . We conjecture that for every connected graph G , which is not isomorphic to K 2 , the multigraph 2 G obtained from G by doubling each edge admits lir ( 2 G ) <= 2 . This concept is closely related to the well known 1-2-3 Conjecture, Local Irregularity Conjecture, (2, 2) Conjecture and other similar problems concerning edge colorings. We show this conjecture holds for graph classes like paths, cycles, wheels, complete graphs, complete k -partite graphs and bipartite graphs. We also prove the general bound for locally irregular chromatic index for all 2-multigraphs using our result for bipartite graphs. (c) 2023 Elsevier Inc. All rights reserved.
Keyword:
Locally irregular graphs and multigraphs
Decomposition of graphs and multigraphs
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