There has been extensive research on cycle lengths in graphs with large minimum degree. In this paper, we obtain several new and tight results in this area. Let G be a graph with minimum degree at least k + 1. We prove that if G is bipartite, then there are k cycles in G whose lengths form an arithmetic progression with common difference two. For general graph G, we show that G contains ⌊k/2⌋ cycles with consecutive even lengths and k − 3 cycles whose lengths form an arithmetic progression with common difference one or two. In addition, if G is 2-connected and non-bipartite, then G contains ⌊k/2⌋ cycles with consecutive odd lengths.Thomassen (1983) made two conjectures on cycle lengths modulo a fixed integer k: (1) every graph with minimum degree at least k + 1 contains cycles of all even lengths modulo k; (2) every 2-connected non-bipartite graph with minimum degree at least k + 1 contains cycles of all lengths modulo k. These two conjectures, if true, are best possible. Our results confirm both conjectures when k is even. And when k is odd, we show that minimum degree at least k + 4 suffices. This improves all previous results in this direction. Moreover, our results derive new upper bounds of the chromatic number in terms of the longest sequence of cycles with consecutive (even or odd) lengths.

We prove that for every graph H, if a graph G has no (odd) H minor, then its vertex set V (G) can be partitioned into three sets X 1 , X 2 , X 3 such that for each i, the subgraph induced on X i has no component of size larger than a function of H and the maximum degree of G. This improves a previous result of Alon, Ding, Oporowski and Vertigan (2003) stating that V (G) can be partitioned into four such sets if G has no H minor. Our theorem generalizes a result of Esperet and Joret (2014), who proved it for graphs embeddable on a fixed surface and asked whether it is true for graphs with no H minor.As a corollary, we prove that for every positive integer t, if a graph G has no K t+1 minor, then its vertex set V (G) can be partitioned into 3t sets X 1 , . . . , X 3t such that for each i, the subgraph induced on X i has no component of size larger than a function of t. This corollary improves a result of Wood (2010), which states that V (G) can be partitioned into ⌈3.5t + 2⌉ such sets.

In this article, we address the maximum number of vertices of induced forests in subcubic graphs with girth at least four or five. We provide a unified approach to prove that every 2‐connected subcubic graph on n vertices and m edges with girth at least four or five, respectively, has an induced forest on at least n−29m or n−15m vertices, respectively, except for finitely many exceptional graphs. Our results improve a result of Liu and Zhao and are tight in the sense that the bounds are attained by infinitely many 2‐connected graphs. Equivalently, we prove that such graphs admit feedback vertex sets with size at most 29m or 15m, respectively. Those exceptional graphs will be explicitly constructed, and our result can be easily modified to drop the 2‐connectivity requirement.