A 2‐factor with two components of a graph satisfying the Chvátal‐Erdös condition

Abstract

Chvá tal and Erdö s showed that a k-connected graph with independence number at most k and order at least three is hamiltonian. In this paper, we show that a graph contains a 2-factor with two components, i.e., the graph can be divided into two cycles if the graph is kð! 4Þ-connected with order at least six and independence number at most k.

The upper bound of the number of cycles in a 2‐factor of a line graph

“…Taking sum of (3), (7), (10), and (11), we obtain |F | + |L| + n − |F | + |L| > n − 6 4 + |F | + 2 + 3γ 0 + γ 1 + 2|L| + 2α − 2γ 0 − γ 1 =⇒ n > n − 6 4 + |F | + 2 + γ 0 + 2α…”

“…There are various results about the number of the components in a 2-factor which is a 2-regular spanning subgraph, see [1,2,7,9,10]. In this article, we study the upper bound of the number of cycles in 2-factors in a line graph.…”

On degree sum conditions for 2-factors with a prescribed number of cycles

“…Chen et al also proved that if the order of a 2-connected graph G with α(G) = α ≤ κ(G) is sufficiently large compared with k and with the Ramsey number r(α + 4, α + 1), then the graph G has a 2-factor with k cycles. In [12], Kaneko and Yoshimoto "almost" solved the above conjecture for k = 2 (see the comment after Theorem E in Chen et al [6] for more details). Another related result can be found in [7].…”

Small cycles and 2-factor passing through any given vertices in graphs

“…Lots of results in this area have been given since 1960s. A related line of research concerning 2-factor is pursued in [8,10,12,13].…”