Structure and coloring of a family of ($P_7, C_5$)-free graphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917736393736192 |
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| author | Chen, Ran Xu, Baogang |
| author_facet | Chen, Ran Xu, Baogang |
| contents | Let $P_t$ and $C_t$ be a path and a cycle on $t$ vertices, respectively. In 2021, Choudum {\em et al.} [Disc. Math. 344 (2021) 112244] determined the structures of $(P_7,C_7,C_4$, diamond)-free and $(P_7,C_7,C_4$, gem)-frees, and gave correspondingly tight upper bounds to the chromatic numbers of these graphs. In this paper, we study the structure of $(P_7, C_5$, kite, paraglider)-free graphs, which is a superfamily of $(P_7, C_5$, diamond)-free graphs. We show that there is a unique connected imperfect $(P_7, C_5$, kite, paraglider)-free graph with $δ(G)\geqω(G)+1$, which has no clique cutsets, no universal cliques, and no pair of vertices of which one's neighborhoods contains the other's. As a consequence, we show that $(P_7, C_5$, kite, paraglider)-free graphs are $χ$-polydet with a binding function $ω(G)+1$. Where a {\em diamond} (resp. {\em gem}) consists of a $P_3$ (resp. $P_4$) and a new vertex adjacent to all vertices of the $P_3$ (resp. $P_4$), a {\em kite} consists of a $P_4$ and a new vertex adjacent to consecutive three vertices of the $P_4$, and a {\em paraglider} consists of a $C_4$ and a new vertex adjacent to three vertices of the $C_4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_17745 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Structure and coloring of a family of ($P_7, C_5$)-free graphs Chen, Ran Xu, Baogang Combinatorics Let $P_t$ and $C_t$ be a path and a cycle on $t$ vertices, respectively. In 2021, Choudum {\em et al.} [Disc. Math. 344 (2021) 112244] determined the structures of $(P_7,C_7,C_4$, diamond)-free and $(P_7,C_7,C_4$, gem)-frees, and gave correspondingly tight upper bounds to the chromatic numbers of these graphs. In this paper, we study the structure of $(P_7, C_5$, kite, paraglider)-free graphs, which is a superfamily of $(P_7, C_5$, diamond)-free graphs. We show that there is a unique connected imperfect $(P_7, C_5$, kite, paraglider)-free graph with $δ(G)\geqω(G)+1$, which has no clique cutsets, no universal cliques, and no pair of vertices of which one's neighborhoods contains the other's. As a consequence, we show that $(P_7, C_5$, kite, paraglider)-free graphs are $χ$-polydet with a binding function $ω(G)+1$. Where a {\em diamond} (resp. {\em gem}) consists of a $P_3$ (resp. $P_4$) and a new vertex adjacent to all vertices of the $P_3$ (resp. $P_4$), a {\em kite} consists of a $P_4$ and a new vertex adjacent to consecutive three vertices of the $P_4$, and a {\em paraglider} consists of a $C_4$ and a new vertex adjacent to three vertices of the $C_4$. |
| title | Structure and coloring of a family of ($P_7, C_5$)-free graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2305.17745 |