The optimal binding function for (cap, even hole)-free graphs
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arXiv
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2025
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| _version_ | 1866908419373400064 |
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| author | Chen, Ran Xu, Baogang Xu, Yian |
| author_facet | Chen, Ran Xu, Baogang Xu, Yian |
| contents | A {\em hole} is an induced cycle of length at least 4, an {\em even hole} is a hole of even length, and a {\em cap} is a graph obtained from a hole by adding an additional vertex which is adjacent exactly to two adjacent vertices of the hole. A graph $G$ obtained from a graph $H$ by blowing up all the vertices into cliques is said to be a clique blowup of $H$. Let $p, q$ be two positive integers with $p>2q$, let $F$ be a triangle-free graph, and let $G'$ be a clique blowup of $F$ with $ω(G')\leq\max\{\frac{2q(p-q-2)}{p-2q}, 2q\}$. In this paper, we prove that for any clique blowup $G$ of $F$, $χ(G)\leq\lceil\frac{p}{2q}ω(G)\rceil$ if and only if $χ(G')\leq\lceil\frac{p}{2q}ω(G')\rceil$. As its consequences, we show that every (cap, even hole)-free graph $G$ satisfies $χ(G)\leq\lceil\frac{5}{4}ω(G)\rceil$, which affirmatively answers a question of Cameron {\em et al.} \cite{CdHV2018}, we also show that every (cap, even hole, 5-hole)-free graph $G$ satisfies $χ(G)\leq\lceil\frac{7}{6}ω(G)\rceil$, and the bound is reachable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_19580 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The optimal binding function for (cap, even hole)-free graphs Chen, Ran Xu, Baogang Xu, Yian Combinatorics A {\em hole} is an induced cycle of length at least 4, an {\em even hole} is a hole of even length, and a {\em cap} is a graph obtained from a hole by adding an additional vertex which is adjacent exactly to two adjacent vertices of the hole. A graph $G$ obtained from a graph $H$ by blowing up all the vertices into cliques is said to be a clique blowup of $H$. Let $p, q$ be two positive integers with $p>2q$, let $F$ be a triangle-free graph, and let $G'$ be a clique blowup of $F$ with $ω(G')\leq\max\{\frac{2q(p-q-2)}{p-2q}, 2q\}$. In this paper, we prove that for any clique blowup $G$ of $F$, $χ(G)\leq\lceil\frac{p}{2q}ω(G)\rceil$ if and only if $χ(G')\leq\lceil\frac{p}{2q}ω(G')\rceil$. As its consequences, we show that every (cap, even hole)-free graph $G$ satisfies $χ(G)\leq\lceil\frac{5}{4}ω(G)\rceil$, which affirmatively answers a question of Cameron {\em et al.} \cite{CdHV2018}, we also show that every (cap, even hole, 5-hole)-free graph $G$ satisfies $χ(G)\leq\lceil\frac{7}{6}ω(G)\rceil$, and the bound is reachable. |
| title | The optimal binding function for (cap, even hole)-free graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2506.19580 |