1.
Tight toughness variant condition for fractional k-factorsWei Gao,
Weifan Wang,
Yaojun Chen, 2026, original scientific article
Abstract: The toughness t(G) of graph G is formalized as the minimum ratio of |S| and ω(G − S) over all vertex subsets S subject to ω(G − S) > 1. As the unique variant parameter of toughness, τ(G) is formulated as the minimum ratio of |S| and ω(G − S) − 1 traversing all the vertex subset S restricted to ω(G − S) ≥ 2. The extant contributions reveal that there is a substantial correlation between toughness and fractional factors. However, there is still a paucity of solid studies on toughness variants τ(G). This work provides several theoretical underpinnings for the tight toughness variant bound for a graph G which admits a fractional k-factor. To be specific, a graph G has a fractional k-factor if τ(G) > k for k ≥ 3 and if τ(G)>3/2 for k = 2. The sharpness of the given bounds is explained by counterexamples.
Keywords: graph, toughness, toughness variant, fractional k-factor
Published in RUP: 21.12.2025; Views: 545; Downloads: 2
Full text (1,11 MB)
