Structure and linear-Pollyanna for some square-free graphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912575051005952 |
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| author | Chen, Ran Xu, Baogang |
| author_facet | Chen, Ran Xu, Baogang |
| contents | We use $P_t$ and $C_t$ to denote a path and a cycle on $t$ vertices, respectively. A {\em bull} is a graph consisting of a triangle with two disjoint pendant edges, a {\em hammer} is a graph obtained by identifying an endvertex of a $P_3$ with a vertex of a triangle. A class ${\cal F}$ is $χ$-bounded if there is a function $f$ such that $χ(G)\leq f(ω(G))$ for all induced subgraphs $G$ of a graph in ${\cal F}$. A class ${\cal C}$ of graphs is {\em Pollyanna} (resp. {\em linear-Pollyanna}) if ${\cal C}\cap {\cal F}$ is polynomially (resp. linear-polynomially) $χ$-bounded for every $χ$-bounded class ${\cal F}$ of graphs. Chudnovsky {\em et al} \cite{CCDO2023} showed that both the classes of bull-free graphs and hammer-free graphs are Pollyannas. Let $G$ be a connected graph with no clique cutsets and no universal cliques. In this paper, we show that $G$ is $(C_4$, hammer)-free if and only if it has girth at least 5, and $G$ is $(C_4$, bull)-free if and only if it is a clique blowup of some graph of girth at least 5. As a consequence, we show that both the classes of $(C_4$, bull)-free graphs and $(C_4$, hammer)-free graphs are linear-Pollyannas. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18506 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Structure and linear-Pollyanna for some square-free graphs Chen, Ran Xu, Baogang Combinatorics We use $P_t$ and $C_t$ to denote a path and a cycle on $t$ vertices, respectively. A {\em bull} is a graph consisting of a triangle with two disjoint pendant edges, a {\em hammer} is a graph obtained by identifying an endvertex of a $P_3$ with a vertex of a triangle. A class ${\cal F}$ is $χ$-bounded if there is a function $f$ such that $χ(G)\leq f(ω(G))$ for all induced subgraphs $G$ of a graph in ${\cal F}$. A class ${\cal C}$ of graphs is {\em Pollyanna} (resp. {\em linear-Pollyanna}) if ${\cal C}\cap {\cal F}$ is polynomially (resp. linear-polynomially) $χ$-bounded for every $χ$-bounded class ${\cal F}$ of graphs. Chudnovsky {\em et al} \cite{CCDO2023} showed that both the classes of bull-free graphs and hammer-free graphs are Pollyannas. Let $G$ be a connected graph with no clique cutsets and no universal cliques. In this paper, we show that $G$ is $(C_4$, hammer)-free if and only if it has girth at least 5, and $G$ is $(C_4$, bull)-free if and only if it is a clique blowup of some graph of girth at least 5. As a consequence, we show that both the classes of $(C_4$, bull)-free graphs and $(C_4$, hammer)-free graphs are linear-Pollyannas. |
| title | Structure and linear-Pollyanna for some square-free graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2407.18506 |