Color-critical Graphs and Hereditary Hypergraphs

TL;DR

This paper presents a new proof of Gallai's theorem on color-critical graphs using hypergraph partitioning, extends the concepts to hereditary hypergraphs, and discusses applications, algorithms, and complexity results.

Contribution

It introduces a novel proof technique for Gallai's theorem based on hypergraph partitions and generalizes the chromatic number to hereditary hypergraphs.

Findings

New proof of Gallai's theorem using hypergraph partitions

Applications to new problems in graph theory

Framework for algorithms and complexity analysis

Abstract

A quick proof of Gallai's celebrated theorem on color-critical graphs is given from Gallai's simple, ingenious lemma on factor-critical graphs, in terms of partitioning the vertex-set into a minimum number of hyperedges of a hereditary hypergraph, generalizing the chromatic number. We then show examples of applying the results to new problems and indicate the way to algorithms and refined complexity results for all these examples at the same time.