A strong k-edge coloring of a graph G is an assignment of k colors to the edges of G such that for any two edges e and f with distance at most two receive different colors. The minimum number of k such that G has a strong k-edge coloring is called the strong chromatic index of G, denoted as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\chi '_s(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>χ</mi> <mi>s</mi> <mo>′</mo> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we proved that for a subquartic graph G, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\chi _s'(G)\le 12$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>χ</mi> <mi>s</mi> <mo>′</mo> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mn>12</mn> </mrow> </math></EquationSource> </InlineEquation> if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$mad(G)<\frac{14}{5}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mi>a</mi> <mi>d</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo><</mo> <mfrac> <mn>14</mn> <mn>5</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$mad(G)=\max \{\frac{2|E(H)|}{|V(H)|},H\subseteq G\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mi>a</mi> <mi>d</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo movablelimits="true">max</mo> <mrow> <mo stretchy="false">{</mo> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">|</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> </mrow> <mrow> <mo stretchy="false">|</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> </mrow> </mfrac> <mo>,</mo> <mi>H</mi> <mo>⊆</mo> <mi>G</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

Strong edge chromatic index of subquartic graphs

期刊论文
发表日期：

2026年6月30日

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Junlei Zhu,
Hongguo Zhu,
Yuehua Bu

摘要

A strong k-edge coloring of a graph G is an assignment of k colors to the edges of G such that for any two edges e and f with distance at most two receive different colors. The minimum number of k such that G has a strong k-edge coloring is called the strong chromatic index of G, denoted as $\chi '_s(G)$ $χ_{s}^{'} (G)$ . In this paper, we proved that for a subquartic graph G, $\chi _s'(G)\le 12$ $χ_{s}^{'} (G) \leq 12$ if $mad(G)<\frac{14}{5}$ $m a d (G) < \frac{14}{5}$ , where $mad(G)=\max \{\frac{2|E(H)|}{|V(H)|},H\subseteq G\}$ $m a d (G) = max {\frac{2 | E (H) |}{| V (H) |}, H \subseteq G}$ .