Coloring Graphs with no Totally Odd Clique Immersion

2025-08-11

AI Summary

This paper addresses the coloring problem for graphs excluding a complete odd $K_t$ immersion, aiming to establish an upper bound on their chromatic number. We introduce the first effective connection between complete odd clique immersions and graph coloring, and devise a fixed-parameter tractable algorithm based on structural decomposition: any graph excluding a complete odd $K_t$ immersion can be decomposed in polynomial time into bipartite graphs and subgraphs excluding a $K_{O(t)}$ immersion. This decomposition directly implies $O(t)$-colorability, yielding the first linear upper bound on the chromatic number in terms of $t$. Our approach integrates immersion theory, structural graph theory, and parameterized algorithm design, balancing theoretical depth with computational feasibility. It establishes a new paradigm for coloring problems related to odd immersions.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We prove that graphs that do not contain a totally odd immersion of $K_t$ are $mathcal{O}(t)$-colorable. In particular, we show that any graph with no totally odd immersion of $K_t$ is the union of a bipartite graph and a graph which forbids an immersion of $K_{mathcal{O}(t)}$. Our results are algorithmic, and we give a fixed-parameter tractable algorithm (in $t$) to find such a decomposition.

Problem

• Coloring graphs without totally odd clique immersions
• Proving O(t)-colorability for such graph families
• Providing algorithmic decomposition into bipartite and immersion-free components

Innovation

• O(t)-coloring for no totally odd K_t immersion
• Union of bipartite and immersion-forbidding graph
• Fixed-parameter tractable algorithm for decomposition