A new improvement to the Overfull Conjecture
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918236453339136 |
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| author | Qi, Xuli Ge, Chunhui Feng, Yanrui |
| author_facet | Qi, Xuli Ge, Chunhui Feng, Yanrui |
| contents | Let $G$ be a simple graph with order $n$, maximum degree $\D(G)$, minimum degree $δ(G)$ and chromatic index $χ'(G)$, respectively. A graph $G$ is called {\em $\D$-critical} if $χ'(G)=\D(G)+1$ and $χ'(H)\textless χ'(G)$ for every proper subgraph $H$ of $G$, and $G$ is overfull if $\left|E(G)\right|>Δ(G)\lfloor n/2\rfloor$. In 1986, Chetwynd and Hilton proposed the Overfull Conjecture: Every $\D$-critical graph $G$ with $\D(G)\textgreater\frac{n}{3}$ is overfull. The Overfull Conjecture has many implications, such as that it implies a polynomial-time algorithm for determining the chromatic index of graphs $G$ with $\D(G)\textgreater\frac{n}{3}$, and implies several longstanding conjectures in the area of graph edge coloring. Recently, Cao, Chen, Jing and Shan (SIAM J. Discrete Math. 2022) verified the Overfull Conjecture for $\D(G)-7δ(G)/4\ge (3n-17)/4$. In this paper, we improve it for $\D(G)-5δ(G)/3\ge (2n-7)/3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_07044 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new improvement to the Overfull Conjecture Qi, Xuli Ge, Chunhui Feng, Yanrui Combinatorics Let $G$ be a simple graph with order $n$, maximum degree $\D(G)$, minimum degree $δ(G)$ and chromatic index $χ'(G)$, respectively. A graph $G$ is called {\em $\D$-critical} if $χ'(G)=\D(G)+1$ and $χ'(H)\textless χ'(G)$ for every proper subgraph $H$ of $G$, and $G$ is overfull if $\left|E(G)\right|>Δ(G)\lfloor n/2\rfloor$. In 1986, Chetwynd and Hilton proposed the Overfull Conjecture: Every $\D$-critical graph $G$ with $\D(G)\textgreater\frac{n}{3}$ is overfull. The Overfull Conjecture has many implications, such as that it implies a polynomial-time algorithm for determining the chromatic index of graphs $G$ with $\D(G)\textgreater\frac{n}{3}$, and implies several longstanding conjectures in the area of graph edge coloring. Recently, Cao, Chen, Jing and Shan (SIAM J. Discrete Math. 2022) verified the Overfull Conjecture for $\D(G)-7δ(G)/4\ge (3n-17)/4$. In this paper, we improve it for $\D(G)-5δ(G)/3\ge (2n-7)/3$. |
| title | A new improvement to the Overfull Conjecture |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.07044 |