Pseudo-multifan and Lollipop
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866914772236107776 |
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| author | Cao, Yan Chen, Guantao Jing, Guangming Shan, Songling |
| author_facet | Cao, Yan Chen, Guantao Jing, Guangming Shan, Songling |
| contents | A simple graph $G$ with maximum degree $Δ$ is \emph{overfull} if $|E(G)|>Δ\lfloor |V(G)|/2\rfloor$. The \emph{core} of $G$, denoted $G_Δ$, is the subgraph of $G$ induced by its vertices of degree $Δ$. Clearly, the chromatic index of $G$ equals $Δ+1$ if $G$ is overfull. Conversely, Hilton and Zhao in 1996 conjectured that if $G$ is a simple connected graph with $Δ\ge 3$ and $Δ(G_Δ)\le 2$, then $χ'(G)=Δ+1$ implies that $G$ is overfull or $G=P^*$, where $P^*$ is obtained from the Petersen graph by deleting a vertex (Core Conjecture). The goal of this paper is to develop the concepts of ``pseudo-multifan'' and ``lollipop'' and study their properties in an edge colored graph. These concepts turn out to be powerful tools in edge coloring graphs with a small core degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_03549 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Pseudo-multifan and Lollipop Cao, Yan Chen, Guantao Jing, Guangming Shan, Songling Combinatorics A simple graph $G$ with maximum degree $Δ$ is \emph{overfull} if $|E(G)|>Δ\lfloor |V(G)|/2\rfloor$. The \emph{core} of $G$, denoted $G_Δ$, is the subgraph of $G$ induced by its vertices of degree $Δ$. Clearly, the chromatic index of $G$ equals $Δ+1$ if $G$ is overfull. Conversely, Hilton and Zhao in 1996 conjectured that if $G$ is a simple connected graph with $Δ\ge 3$ and $Δ(G_Δ)\le 2$, then $χ'(G)=Δ+1$ implies that $G$ is overfull or $G=P^*$, where $P^*$ is obtained from the Petersen graph by deleting a vertex (Core Conjecture). The goal of this paper is to develop the concepts of ``pseudo-multifan'' and ``lollipop'' and study their properties in an edge colored graph. These concepts turn out to be powerful tools in edge coloring graphs with a small core degree. |
| title | Pseudo-multifan and Lollipop |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2108.03549 |