Immersions of large cliques in graphs with independence number 2 and bounded maximum degree
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| Formato: | Preprint |
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2025
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| author | Botler, Fábio Fernandes, Cristina G. Lintzmayer, Carla N. Lopes, Rui A. Mishra, Suchismita Netto, Bruno L. Sambinelli, Maycon |
| author_facet | Botler, Fábio Fernandes, Cristina G. Lintzmayer, Carla N. Lopes, Rui A. Mishra, Suchismita Netto, Bruno L. Sambinelli, Maycon |
| contents | An immersion of a graph $H$ in a graph $G$ is a minimal subgraph $I$ of $G$ for which there is an injection ${\rm i} \colon V(H) \to V(I)$ and a set of edge-disjoint paths $\{P_e: e \in E(H)\}$ in $I$ such that the end vertices of $P_{uv}$ are precisely ${\rm i}(u)$ and ${\rm i}(v)$. The immersion analogue of Hadwiger Conjecture (1943), posed by Lescure and Meyniel (1985), asks whether every graph $G$ contains an immersion of $K_{χ(G)}$. Its restriction to graphs with independence number 2 has received some attention recently, and Vergara (2017) raised the weaker conjecture that every graph with independence number 2 has an immersion of $K_{χ(G)}$. This implies that every graph with independence number 2 has an immersion of $K_{\lceil n/2 \rceil}$. In this paper, we verify Vergara Conjecture for graphs with bounded maximum degree. Specifically, we prove that if $G$ is a graph with independence number $2$, maximum degree less than $2n/3 - 1$ and clique covering number at most $3$, then $G$ contains an immersion of $K_{χ(G)}$ (and thus of $K_{\lceil n/2 \rceil}$). Using a result of Jin (1995), this implies that if $G$ is a graph with independence number $2$ and maximum degree less than $19n/29 - 1$, then $G$ contains an immersion of $K_{χ(G)}$ (and thus of $K_{\lceil n/2 \rceil}$). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_09768 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Immersions of large cliques in graphs with independence number 2 and bounded maximum degree Botler, Fábio Fernandes, Cristina G. Lintzmayer, Carla N. Lopes, Rui A. Mishra, Suchismita Netto, Bruno L. Sambinelli, Maycon Combinatorics Discrete Mathematics An immersion of a graph $H$ in a graph $G$ is a minimal subgraph $I$ of $G$ for which there is an injection ${\rm i} \colon V(H) \to V(I)$ and a set of edge-disjoint paths $\{P_e: e \in E(H)\}$ in $I$ such that the end vertices of $P_{uv}$ are precisely ${\rm i}(u)$ and ${\rm i}(v)$. The immersion analogue of Hadwiger Conjecture (1943), posed by Lescure and Meyniel (1985), asks whether every graph $G$ contains an immersion of $K_{χ(G)}$. Its restriction to graphs with independence number 2 has received some attention recently, and Vergara (2017) raised the weaker conjecture that every graph with independence number 2 has an immersion of $K_{χ(G)}$. This implies that every graph with independence number 2 has an immersion of $K_{\lceil n/2 \rceil}$. In this paper, we verify Vergara Conjecture for graphs with bounded maximum degree. Specifically, we prove that if $G$ is a graph with independence number $2$, maximum degree less than $2n/3 - 1$ and clique covering number at most $3$, then $G$ contains an immersion of $K_{χ(G)}$ (and thus of $K_{\lceil n/2 \rceil}$). Using a result of Jin (1995), this implies that if $G$ is a graph with independence number $2$ and maximum degree less than $19n/29 - 1$, then $G$ contains an immersion of $K_{χ(G)}$ (and thus of $K_{\lceil n/2 \rceil}$). |
| title | Immersions of large cliques in graphs with independence number 2 and bounded maximum degree |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2506.09768 |