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

Xiangzhi Tu (), Peng Li (), Yangjing Long () and Aifa Wang ()
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Xiangzhi Tu: Chongqing University of Technology
Peng Li: Chongqing University of Technology
Yangjing Long: Central China Normal University
Aifa Wang: Chongqing University of Technology

Journal of Combinatorial Optimization, 2025, vol. 49, issue 4, No 15, 12 pages

Abstract: Abstract 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 , ⋯ , k . If the condition $$S\left( u \right) \ne S\left( v \right) $$ S u ≠ S v holds for any edge $$uv\in E\left( G \right) $$ u v ∈ E G , where $$S\left( u \right) =\sum \nolimits _{uv\in E\left( G \right) }{\sigma \left( uv \right) }$$ S u = ∑ u v ∈ 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 .

Keywords: Joint graphs; Coloring matrices; Neighbor sum distinguishable edge coloring; Neighbor sum distinguishable edge coloring numbers (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10878-025-01309-z

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