Coloring Graphs With No Totally Odd Clique Immersion

TL;DR

This paper proves that graphs without a totally odd immersion of a complete graph are colorable with a number of colors proportional to the size of the immersion, providing a decomposition and an efficient algorithm.

Contribution

It establishes a linear bound on the chromatic number for graphs excluding a totally odd immersion of $K_t$ and offers a fixed-parameter tractable algorithm for decomposition.

Findings

Graphs with no totally odd $K_t$ immersion are $igO(t)$-colorable.

Such graphs can be decomposed into a bipartite graph and a graph forbidding a $K_{igO(t)}$ immersion.

Provides an algorithmic, fixed-parameter tractable method for finding the decomposition.

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.