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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.01314 |
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| _version_ | 1866913570531311616 |
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| author | Couto, Fernanda Ferraz, Diego Amaro Klein, Sulamita |
| author_facet | Couto, Fernanda Ferraz, Diego Amaro Klein, Sulamita |
| contents | A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. A connected graph $G$ is said to be $t$-admissible if admits a spanning tree in which the distance between any two adjacent vertices of $G$ is at most $t$. Given a graph $G$, determining the smallest $t$ for which $G$ is $t$-admissible, i.e., the stretch index of $G$ denoted by $σ(G)$, is the goal of the $t$-admissibility problem. Split graphs are $3$-admissible and can be partitioned into three subclasses: split graphs with $σ= 1$, $2$ or $3$. In this work we consider such a partition while dealing with the problem of coloring the edges of a split graph. Vizing proved that any graph can have its edges colored with $Δ$ or $Δ+1$ colors, and thus can be classified as Class $1$ or Class $2$, respectively. The edge coloring problem is open for split graphs in general. In previous results, we classified split graphs with $σ= 2$ and in this paper we classify and provide an algorithm to color the edges of a subclass of split graphs with $σ= 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_01314 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Filling some gaps on the edge coloring problem of split graphs Couto, Fernanda Ferraz, Diego Amaro Klein, Sulamita Combinatorics Discrete Mathematics A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. A connected graph $G$ is said to be $t$-admissible if admits a spanning tree in which the distance between any two adjacent vertices of $G$ is at most $t$. Given a graph $G$, determining the smallest $t$ for which $G$ is $t$-admissible, i.e., the stretch index of $G$ denoted by $σ(G)$, is the goal of the $t$-admissibility problem. Split graphs are $3$-admissible and can be partitioned into three subclasses: split graphs with $σ= 1$, $2$ or $3$. In this work we consider such a partition while dealing with the problem of coloring the edges of a split graph. Vizing proved that any graph can have its edges colored with $Δ$ or $Δ+1$ colors, and thus can be classified as Class $1$ or Class $2$, respectively. The edge coloring problem is open for split graphs in general. In previous results, we classified split graphs with $σ= 2$ and in this paper we classify and provide an algorithm to color the edges of a subclass of split graphs with $σ= 3$. |
| title | Filling some gaps on the edge coloring problem of split graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2411.01314 |