Connected matching in graphs with independence number two
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912019367591936 |
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| author | Chen, Rong Deng, Zijian |
| author_facet | Chen, Rong Deng, Zijian |
| contents | A matching $M$ in a graph $G$ is {\em connected} if $G$ has an edge linking each pair of edges in $M$. The problem to find large connected matchings in graphs $G$ with $α(G)=2$ is closely related to Hadwiger's conjecture for graphs with independence number 2. The problem of finding a large connected matching in a general graph is NP-hard. F{ü}redi et al. in 2005 conjectured that each $(4t-1)$-vertex graph $G$ with $α(G)=2$ contains a connected matching of size at least $t$. Cambie recently showed that if this conjecture is false, then so is Hadwiger's conjecture. In this paper, we present a number of properties possessed by a counterexample to F{ü}redi et al.'s conjecture, and then using these properties, we prove that F{ü}redi et al.'s conjecture holds for $t\leq22$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05920 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Connected matching in graphs with independence number two Chen, Rong Deng, Zijian Combinatorics A matching $M$ in a graph $G$ is {\em connected} if $G$ has an edge linking each pair of edges in $M$. The problem to find large connected matchings in graphs $G$ with $α(G)=2$ is closely related to Hadwiger's conjecture for graphs with independence number 2. The problem of finding a large connected matching in a general graph is NP-hard. F{ü}redi et al. in 2005 conjectured that each $(4t-1)$-vertex graph $G$ with $α(G)=2$ contains a connected matching of size at least $t$. Cambie recently showed that if this conjecture is false, then so is Hadwiger's conjecture. In this paper, we present a number of properties possessed by a counterexample to F{ü}redi et al.'s conjecture, and then using these properties, we prove that F{ü}redi et al.'s conjecture holds for $t\leq22$. |
| title | Connected matching in graphs with independence number two |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.05920 |