Coloring Graphs With No Totally Odd Clique Immersion
Caleb McFarland

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 are -colorable. In particular, we show that any graph with no totally odd immersion of is the union of a bipartite graph and a graph which forbids an immersion of . Our results are algorithmic, and we give a fixed-parameter tractable algorithm (in ) to find such a decomposition.
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Taxonomy
TopicsAdvanced Graph Theory Research · Computational Geometry and Mesh Generation · Topological and Geometric Data Analysis
