• Corpus ID: 102352037

Minimum degree conditions for the existence of cycles of all lengths modulo $k$ in graphs

@article{Chiba2019MinimumDC,
  title={Minimum degree conditions for the existence of cycles of all lengths modulo \$k\$ in graphs},
  author={Shuya Chiba and Tomoki Yamashita},
  journal={arXiv: Combinatorics},
  year={2019}
}
Thomassen, in 1983, conjectured that for a positive integer $k$, every $2$-connected non-bipartite graph of minimum degree at least $k + 1$ contains cycles of all lengths modulo $k$. In this paper, we settle this conjecture affirmatively. 

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TLDR
The existence of a function α(k) (where k is a natural number) is established such that the vertex set of any graph G has a decomposition A ∪ B ∪ C such that G has minimum degree at least k.

By adding a (b, y)-path in G − (V (C) ∪ V (B − b)) to each of the l − 1 (x, b)-paths, we can get l − 1 (x, y)-paths in G − V (C) satisfying the length condition

  • We next consider the case where (ii) holds. Then it follows from (6.1)

Repeating this argument we get N G (C) ⊆ V (B − b) since |V (C)| is odd. This implies that b is a cut vertex of G

    B − b) = ∅. Then again by applying (i) of Claim 6.3 for the vertex u +m and a neighbor of u +m in B − b instead of u and x, respectively