Edge-critical subgraphs of Schrijver graphs

Abstract

For k ≥ 1 and n ≥ 2k, the Kneser graph KG(n, k) has all k-element subsets of an n-element set as vertices; two such subsets are adjacent if they are disjoint. It was first proved by Lovász that the chromatic number of KG(n, k) is n − 2k + 2. Schrijver constructed a vertex-critical subgraph SG(n, k) of KG(n, k) with the same chromatic number. For the stronger notion of criticality defined in terms of removing edges, however, no analogous construction is known except in trivial cases. We provide such a construction for k = 2 and arbitrary n ≥ 4 by means of a nice explicit combinatorial definition. arXiv:1910.07866v1 [math.CO]

Colorings of complements of line graphs

“…When , this is the special case of our result when there is exactly one cycle in the collection. (The question of edge‐criticality is probably trickier; see the recent work by Matěj Stehlík and Tomáš Kaiser [13], who describe a family of edge‐critical spanning subgraphs of . )…”

Edge-critical subgraphs of Schrijver graphs II: The general case

“…). In the paper [5], pairs of these two types are referred to as crossing and transverse pairs, respectively, and they coincide with the edges of the graph studied in that paper (denoted by G n ). Thus, as noted in Section 1, the present definition specialises to the one of [5] for k = 2.…”

“…We remark that the definition of almost-interlacing edges is particularly simple for the case k = 2. Indeed, almost-interlacing edges of SG(n, 2) correspond to crossing and transverse edges defined in [5], so the graph XG(n, 2) is precisely the graph G n studied in [5].…”

Colorings of complements of line graphs

“…When k = 2, this is the special case of our result when there is exactly one cycle in the collection. (The question of edge-criticality is probably trickier; see the recent work by Matěj Stehlík and Tomáš Kaiser [12], who describe a family of edge-critical spanning subgraphs of SG(n, 2). )…”