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<p>A strong edge-coloring of a graph <i>G</i> is an edge-coloring such that any two edges of distance at most two receive distinct colors. The minimum number of colors we need in order to give <i>G</i> a strong edge-coloring is called the strong chromatic index of <i>G</i>, denoted by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\chi _s'(G)\)</EquationSource> </InlineEquation>. The maximum edge weight of <i>G</i> is defined to be <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\max \{d(u)+d(v):\ uv\in E(G)\}\)</EquationSource> </InlineEquation>. In this paper, using the discharging method, we prove that if <i>G</i> is a graph with maximum edge weight 7 and maximum average degree less than <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\frac{40}{13}\)</EquationSource> </InlineEquation>, then <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\chi _s'(G)\le 13\)</EquationSource> </InlineEquation>. Also, we determine the largest possible maximum average degree of a graph with given maximum edge weight.</p>

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Strong edge-coloring of graphs with maximum edge weight seven

  • 期刊论文

  • 发表日期：

    2025年12月21日

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  • Runze Wang

摘要

A strong edge-coloring of a graph G is an edge-coloring such that any two edges of distance at most two receive distinct colors. The minimum number of colors we need in order to give G a strong edge-coloring is called the strong chromatic index of G, denoted by \(\chi _s'(G)\) . The maximum edge weight of G is defined to be \(\max \{d(u)+d(v):\ uv\in E(G)\}\) . In this paper, using the discharging method, we prove that if G is a graph with maximum edge weight 7 and maximum average degree less than \(\frac{40}{13}\) , then \(\chi _s'(G)\le 13\) . Also, we determine the largest possible maximum average degree of a graph with given maximum edge weight.