Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.00929 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908512092684288 |
|---|---|
| author | Ji, Yuqing Wang, Yue Yang, Yujun Zhang, Xia |
| author_facet | Ji, Yuqing Wang, Yue Yang, Yujun Zhang, Xia |
| contents | A paraglider, house, 4-wheel, is the graph that consists of a cycle $C_4$ plus an additional vertex adjacent to three vertices, two adjacent vertices, all the vertices of the $C_4$, respectively. For a graph $G$, let $χ(G)$, $ω(G)$ denote the chromatic number, the clique number of $G$, respectively. Gerards and Seymour from 1995 conjectured that every graph $G$ has an odd $K_{χ(G)}$ minor. In this paper, based on the description of graph structure, it is shown that every graph $G$ with independence number two satisfies the conjecture if one of the following is true: $χ(G) \leq 2ω(G)$ when $n $ is even, $χ(G) \leq 9ω(G)/5$ when $n$ is odd, $G$ is a quasi-line graph, $G$ is $H$-free for some induced subgraph $H$ of paraglider, house or $W_4$. Moreover, we derive an optimal linear $χ$-binding function for {3$K_1$, paraglider}-free graph $G$ that $χ(G)\leq \max\{ω(G)+3, 2ω(G)-2\}$, which improves the previous result, $χ(G)\leq 2ω(G)$, due to Choudum, Karthick and Shalu in 2008. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_00929 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Odd clique minors and chromatic bounds of {3$K_1$, paraglider}-free graphs Ji, Yuqing Wang, Yue Yang, Yujun Zhang, Xia Combinatorics A paraglider, house, 4-wheel, is the graph that consists of a cycle $C_4$ plus an additional vertex adjacent to three vertices, two adjacent vertices, all the vertices of the $C_4$, respectively. For a graph $G$, let $χ(G)$, $ω(G)$ denote the chromatic number, the clique number of $G$, respectively. Gerards and Seymour from 1995 conjectured that every graph $G$ has an odd $K_{χ(G)}$ minor. In this paper, based on the description of graph structure, it is shown that every graph $G$ with independence number two satisfies the conjecture if one of the following is true: $χ(G) \leq 2ω(G)$ when $n $ is even, $χ(G) \leq 9ω(G)/5$ when $n$ is odd, $G$ is a quasi-line graph, $G$ is $H$-free for some induced subgraph $H$ of paraglider, house or $W_4$. Moreover, we derive an optimal linear $χ$-binding function for {3$K_1$, paraglider}-free graph $G$ that $χ(G)\leq \max\{ω(G)+3, 2ω(G)-2\}$, which improves the previous result, $χ(G)\leq 2ω(G)$, due to Choudum, Karthick and Shalu in 2008. |
| title | Odd clique minors and chromatic bounds of {3$K_1$, paraglider}-free graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.00929 |