Vertex-distinguishing and sum-distinguishing edge coloring of regular graphs
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910737342922752 |
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| author | Gao, Yuping Shan, Songling Wang, Guanghui |
| author_facet | Gao, Yuping Shan, Songling Wang, Guanghui |
| contents | Given an integer $k\ge1$, an edge-$k$-coloring of a graph $G$ is an assignment of $k$ colors $1,\ldots,k$ to the edges of $G$ such that no two adjacent edges receive the same color. A vertex-distinguishing (resp. sum-distinguishing) edge-$k$-coloring of $G$ is an edge-$k$-coloring such that for any two distinct vertices $u$ and $v$, the set (resp. sum) of colors taken from all the edges incident with $u$ is different from that taken from all the edges incident with $v$. The vertex-distinguishing chromatic index (resp. sum-distinguishing chromatic index), denoted $χ'_{vd}(G)$ (resp. $χ'_{sd}(G)$), is the smallest value $k$ such that $G$ has a vertex-distinguishing-edge-$k$-coloring (resp. sum-distinguishing-edge-$k$-coloring). Let $G$ be a $d$-regular graph on $n$ vertices, where $n$ is even and sufficiently large. We show that $χ'_{vd}(G) =d+2$ if $d$ is arbitrarily close to $n/2$ from above, and $χ'_{sd}(G) =d+2$ if $d\ge \frac{2n}{3}$. Our first result strengthens a result of Balister et al. in 2004 for such class of regular graphs, and our second result constitutes a significant advancement in the field of sum-distinguishing edge coloring.
To achieve these results, we introduce novel edge coloring results which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05352 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Vertex-distinguishing and sum-distinguishing edge coloring of regular graphs Gao, Yuping Shan, Songling Wang, Guanghui Combinatorics Given an integer $k\ge1$, an edge-$k$-coloring of a graph $G$ is an assignment of $k$ colors $1,\ldots,k$ to the edges of $G$ such that no two adjacent edges receive the same color. A vertex-distinguishing (resp. sum-distinguishing) edge-$k$-coloring of $G$ is an edge-$k$-coloring such that for any two distinct vertices $u$ and $v$, the set (resp. sum) of colors taken from all the edges incident with $u$ is different from that taken from all the edges incident with $v$. The vertex-distinguishing chromatic index (resp. sum-distinguishing chromatic index), denoted $χ'_{vd}(G)$ (resp. $χ'_{sd}(G)$), is the smallest value $k$ such that $G$ has a vertex-distinguishing-edge-$k$-coloring (resp. sum-distinguishing-edge-$k$-coloring). Let $G$ be a $d$-regular graph on $n$ vertices, where $n$ is even and sufficiently large. We show that $χ'_{vd}(G) =d+2$ if $d$ is arbitrarily close to $n/2$ from above, and $χ'_{sd}(G) =d+2$ if $d\ge \frac{2n}{3}$. Our first result strengthens a result of Balister et al. in 2004 for such class of regular graphs, and our second result constitutes a significant advancement in the field of sum-distinguishing edge coloring. To achieve these results, we introduce novel edge coloring results which may be of independent interest. |
| title | Vertex-distinguishing and sum-distinguishing edge coloring of regular graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2412.05352 |