Topological Approaches to Chromatic Number and Box Complex Analysis of Partition Graphs

Abstract

Determining the chromatic number of the partition graph P(33) poses a considerable challenge. We can bound it to 4 ≤ χ(P(33)) ≤ 6, with exhaustive search confirming χ(P(33)) = 6. A potential mathematical proof strategy for this equality involves identifying a Z2-invariant S4 with non-trivial homology in the box complex of the partition graph P(33), namely Bedge(︁P(33))︁, and applying the Borsuk-Ulam theorem to compute its Z2-index. This provides a robust topological lower bound for the chromatic number of P(33), termed the Lovász bound. We have verified the absence of such an S4 within certain sections of Bedge(︁P(33))︁. We also validated this approach through a case study on the Petersen graph. This thesis offers a thorough examination of various topological lower bounds for a graph’s chromatic number, complete with proofs and examples. We demonstrate instances where these lower bounds converge to a single value and others where they diverge significantly from a graph’s actual chromatic number. We also classify all vertex pairs, triples, and quadruples of P(33) into unique equivalence classes, facilitating the derivation of all maximal complete bipartite subgraphs. This classification informs the construction of all simplices of Bedge(︁P(33)). Following a detailed and technical exploration, we uncover both the maximal size of the pairwise intersections of its maximal simplices and their underlying structure. Our study proposes an algorithm for building the box complex of the partition graph P(33) using our method of identifying maximal complete bipartite subgraphs. This reduces time complexity to O(n3), marking a substantial enhancement over brute-force techniques. Lastly, we apply discrete Morse theory to construct a simplicial complex homotopy equivalent to the box complex of P(33), using two methods: elementary collapses and the determination of a discrete Morse function on the box complex. This process reduces the dimension of the box complex from 35 to 12, streamlining future calculations of the Z2-index and the Lovász bound.