In a graph G, the normal k-edge coloring <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10878_2025_1309_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> is defined as the conventional edge coloring of G using the color set <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10878_2025_1309_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="136" /> </InlineMediaObject> <EquationSource Format="TEX">$\left[ k \right] =\left\{ 1,2,\cdots ,k \right\} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close="]" open="["> <mi>k</mi> </mfenced> <mo>=</mo> <mfenced close="}" open="{"> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>k</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. If the condition <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10878_2025_1309_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="95" /> </InlineMediaObject> <EquationSource Format="TEX">$S\left( u \right) \ne S\left( v \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mfenced close=")" open="("> <mi>u</mi> </mfenced> <mo>≠</mo> <mi>S</mi> <mfenced close=")" open="("> <mi>v</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> holds for any edge <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10878_2025_1309_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$uv\in E\left( G \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mi>v</mi> <mo>∈</mo> <mi>E</mi> <mfenced close=")" open="("> <mi>G</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10878_2025_1309_Article_IEq8.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="173" /> </InlineMediaObject> <EquationSource Format="TEX">$S\left( u \right) =\sum \nolimits _{uv\in E\left( G \right) }{\sigma \left( uv \right) }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mfenced close=")" open="("> <mi>u</mi> </mfenced> <mo>=</mo> <msub> <mo>∑</mo> <mrow> <mi>u</mi> <mi>v</mi> <mo>∈</mo> <mi>E</mi> <mfenced close=")" open="("> <mi>G</mi> </mfenced> </mrow> </msub> <mrow> <mi>σ</mi> <mfenced close=")" open="("> <mi>u</mi> <mi>v</mi> </mfenced> </mrow> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10878_2025_1309_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> is termed a neighbor sum distinguishable k-edge coloring of the graph G, abbreviated as k-VSDEC. The minimum number of colors <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10878_2025_1309_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$ k $</EquationSource> <EquationSource Format="MATHML"><math> <mi>k</mi> </math></EquationSource> </InlineEquation> needed for this type of coloring is referred to as the neighbor sum distinguishable edge chromatic number of <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10878_2025_1309_Article_IEq11.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$ G $</EquationSource> <EquationSource Format="MATHML"><math> <mi>G</mi> </math></EquationSource> </InlineEquation>, represented as <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10878_2025_1309_Article_IEq12.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$ \chi '_{\varSigma }(G) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>χ</mi> <mi>Σ</mi> <mo>′</mo> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. This paper examines neighbor sum distinguishable k-edge colorings in the joint graphs of an h-order path <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10878_2025_1309_Article_IEq13.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">${{P}_{h}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>h</mi> </msub> </math></EquationSource> </InlineEquation> and an <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10878_2025_1309_Article_IEq14.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$\left( z+1 \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mi>z</mi> <mo>+</mo> <mn>1</mn> </mfenced> </math></EquationSource> </InlineEquation>-order star <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10878_2025_1309_Article_IEq15.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">${{S}_{z}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mi>z</mi> </msub> </math></EquationSource> </InlineEquation>, providing exact values for their neighboring and distinguishable edge coloring numbers, which are either <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10878_2025_1309_Article_IEq16.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\varDelta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>Δ</mi> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10878_2025_1309_Article_IEq17.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$\varDelta +1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Δ</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.

Neighbor sum distinguishable $k$ -edge colorings of joint graphs

期刊论文
发表日期：

2025年5月15日

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Xiangzhi Tu,
Peng Li,
Yangjing Long,
Aifa Wang

摘要

In a graph G, the normal k-edge coloring $\sigma $ $σ$ is defined as the conventional edge coloring of G using the color set $\left[ k \right] =\left\{ 1,2,\cdots ,k \right\} $ $[k] = \{1, 2, \dots, k\}$ . If the condition $S\left( u \right) \ne S\left( v \right) $ $S (u) \neq S (v)$ holds for any edge $uv\in E\left( G \right) $ $u v \in E (G)$ , where $S\left( u \right) =\sum \nolimits _{uv\in E\left( G \right) }{\sigma \left( uv \right) }$ $S (u) = \sum_{u v \in E (G)} σ (u, v)$ , then $\sigma $ $σ$ is termed a neighbor sum distinguishable k-edge coloring of the graph G, abbreviated as k-VSDEC. The minimum number of colors $ k $ $k$ needed for this type of coloring is referred to as the neighbor sum distinguishable edge chromatic number of $ G $ $G$ , represented as $ \chi '_{\varSigma }(G) $ $χ_{Σ}^{'} (G)$ . This paper examines neighbor sum distinguishable k-edge colorings in the joint graphs of an h-order path ${{P}_{h}}$ $P_{h}$ and an $\left( z+1 \right) $ $(z + 1)$ -order star ${{S}_{z}}$ $S_{z}$ , providing exact values for their neighboring and distinguishable edge coloring numbers, which are either $\varDelta $ $Δ$ or $\varDelta +1$ $Δ + 1$ .

Neighbor sum distinguishable \(k\) -edge colorings of joint graphs

摘要