Immersions of large cliques in graphs with independence number 2 and bounded maximum degree

2025-06-11

AI Summary

This paper investigates the validity of Vergara’s conjecture—which posits that every graph (G) with independence number (alpha(G) = 2) contains a (K_{chi(G)}) immersion—for graphs with bounded maximum degree (Delta(G)). Method: Integrating immersion theory, chromatic analysis, and clique covering decomposition, the authors combine Jin’s classical result with constraints on the clique covering number to derive new structural bounds. Contribution/Results: The work extends verification of Vergara’s conjecture to the broad class of bounded-degree graphs for the first time. It establishes the strongest degree condition to date: if (Delta < frac{19n}{29} - 1), then (G) admits a (K_{chi(G)}) immersion. Moreover, if the clique covering number is at most 3 and (Delta < frac{2n}{3} - 1), the conclusion also holds. These results significantly advance the theoretical understanding of the interplay between graph immersion and structural parameters such as independence number, chromatic number, maximum degree, and clique cover.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Models for Web evolution

Abstract

An immersion of a graph $H$ in a graph $G$ is a minimal subgraph $I$ of $G$ for which there is an injection ${{ m i}} colon V(H) o V(I)$ 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 ${{ m i}}(u)$ and ${{ m i}}(v)$. The immersion analogue of Hadwiger Conjecture (1943), posed by Lescure and Meyniel (1985), asks whether every graph $G$ contains an immersion of $K_{chi(G)}$. 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 $K_{chi(G)}$. This implies that every graph with independence number 2 has an immersion of $K_{lceil n/2 ceil}$. 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 $K_{chi(G)}$ (and thus of $K_{lceil n/2 ceil}$). 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 $K_{chi(G)}$ (and thus of $K_{lceil n/2 ceil}$).

Problem

• Study immersions of large cliques in graphs with independence number 2.
• Verify Vergara Conjecture for graphs with bounded maximum degree.
• Prove immersion existence under specific degree and clique conditions.

Innovation

• Graph immersion with bounded maximum degree
• Clique immersion in independence-2 graphs
• Maximum degree constraints for immersion