A note on degree sum conditions for 2-factors with a prescribed number of cycles in bipartite graphs

Let $G$ be a balanced bipartite graph of order $2n\ge 4$, and let ${\sigma }_{1,1}\left(G\right)$ be the minimum degree sum of two non-adjacent vertices in different partite sets of $G$. In 1963, Moon and Moser proved that if ${\sigma }_{1,1}\left(G\right)\ge n+1$, then $G$ is hamiltonian. In this note, we show that if $k$ is a positive integer, then the Moon–Moser condition also implies the existence of a 2-factor with exactly $k$ cycles for sufficiently large graphs. In order to prove this, we also give a ${\sigma }_{1,1}$ condition for the existence of $k$ vertex-disjoint alternating cycles with respect to a chosen perfect matching in $G$.     

On vertex-disjoint cycles and degree sum conditions

This paper considers a degree sum condition sufficient to imply the existence of $k$ vertex-disjoint cycles in a graph $G$. For an integer $t\ge 1$, let ${\sigma }_t\left(G\right)$ be the smallest sum of degrees of $t$ independent vertices of $G$. We prove that if $G$ has order at least $7k+1$ and ${\sigma }_4\left(G\right)\ge 8k?3$, with $k\ge 2$, then $G$ contains $k$ vertex-disjoint cycles. We also show that the degree sum condition on ${\sigma }_4\left(G\right)$ is sharp and conjecture a degree sum condition on ${\sigma }_t\left(G\right)$ sufficient to imply $G$ contains $k$ vertex-disjoint cycles for $k\ge 2$.     

On degree sum conditions for long cycles and cycles through specified vertices

Let G be a graph. For S⊂V(G), let Δ_k(S) denote the maximum value of the degree sums of the subsets of S of order k. In this paper, we prove the following two results. (1) Let G be a 2-connected graph. If Δ₂(S)≥d for every independent set S of order κ(G)+1, then G has a cycle of length at least min{d,|V(G)|}. (2) Let G be a 2-connected graph and X a subset of V(G). If Δ₂(S)≥|V(G)| for every independent set S of order κ(X)+1 in G[X], then G has a cycle that includes every vertex of X. This suggests that the degree sum of nonadjacent two vertices is important for guaranteeing the existence of these cycles.     

Degree conditions for the partition of a graph into cycles, edges and isolated vertices

Let k,n be integers with 2≤k≤n, and let G be a graph of order n. We prove that if max{d_G(x),d_G(y)}≥(n−k+1)/2 for any x,y∈V(G) with x≠y and xy∉E(G), then G has k vertex-disjoint subgraphs H₁,…,H_k such that V(H₁)∪?∪V(H_k)=V(G) and H_i is a cycle or K₁ or K₂ for each 1≤i≤k, unless k=2 and G=C₅, or k=3 and G=K₁∪C₅.     

Ore-type conditions implying 2-factors consisting of short cycles

For every graph G, let . The main result of the paper says that every n-vertex graph G with contains each spanning subgraph H all whose components are isomorphic to graphs in . This generalizes the earlier results of Justesen, Enomoto, and Wang, and is a step towards an Ore-type analogue of the Bollobás-Eldridge-Catlin Conjecture.     

disjoint cycles containing specified independent vertices

Let k1 be an integer and G be a graph of order n3k satisfying the condition that σ₂(G)n+k-1. Let v₁,…,v_k be k independent vertices of G, and suppose that G has k vertex-disjoint triangles C₁,…,C_k with v_iV(C_i) for all 1ik.Then G has k vertex-disjoint cycles such that

(i) for all 1ik.

(ii) , and

(iii) At least k-1 of the k cycles are triangles.

The condition of degree sum σ₂(G)n+k-1 is sharp.

Hamilton cycles in circulant digraphs with prescribed number of distinct jumps

Let G be a digraph that consists of a finite set of vertices V(G). G is called a circulant digraph if its automorphism group contains a |V(G)|-cycle. A circulant digraph G has arcs incident to each vertex i, where integers a_ks satisfy 0<a₁<a₂<a_j≤|V(G)|−1 and are called jumps. It is well known that not every G is Hamiltonian. In this paper we give sufficient conditions for a G to have a Hamilton cycle with prescribed distinct jumps, and prove that such conditions are also necessary for every G with two distinct jumps. Finally, we derive several results for obtaining G^′ with k, k≥2 distinct jumps if the corresponding G contains a Hamilton cycle with two distinct jumps.     

Degree sum and connectivity conditions for dominating cycles

For a graph G, let σ_k(G) be the minimum degree sum of an independent set of k vertices. Ore showed that if G is a graph of order n?3 with σ₂(G)?n then G is hamiltonian. Let κ(G) be the connectivity of a graph G. Bauer, Broersma, Li and Veldman proved that if G is a 2-connected graph on n vertices with σ₃(G)?n+κ(G), then G is hamiltonian. On the other hand, Bondy showed that if G is a 2-connected graph on n vertices with σ₃(G)?n+2, then each longest cycle of G is a dominating cycle. In this paper, we prove that if G is a 3-connected graph on n vertices with σ₄(G)?n+κ(G)+3, then G contains a longest cycle which is a dominating cycle.     

A degree sum condition for long cycles passing through a linear forest

Let G be a (k+m)-connected graph and F be a linear forest in G such that |E(F)|=m and F has at most k-2 components of order 1, where k?2 and m?0. In this paper, we prove that if every independent set S of G with |S|=k+1 contains two vertices whose degree sum is at least d, then G has a cycle C of length at least min{d-m,|V(G)|} which contains all the vertices and edges of F.     

Vertex-disjoint directed cycles of prescribed length in tournaments with given minimum out-degree and in-degree

In a recent paper, Bessy, Sereni and the author (see [3]) have proved that for r≥1, a tournament with minimum out-degree and in-degree both greater than or equal to 2r−1 contains at least r vertex-disjoint directed triangles. In this paper, we generalize this result; more precisely, we prove that for given integers q≥3 and r≥1, a tournament with minimum out-degree and in-degree both greater than or equal to (q−1)r−1 contains at least r vertex-disjoint directed cycles of length q. We will use an auxiliary result established in [3], concerning a union of sets contained in another union of sets. We finish by giving a lower bound on the maximum number of vertex-disjoint directed cycles of length q when only the minimum out-degree is supposed to be greater than or equal to (q−1)r−1.     

Degree sum conditions for oriented forests in digraphs

Let F be an oriented forest with n vertices and m arcs and D be a digraph without loops and multiple arcs. In this note we prove that D contains a subdigraph isomorphic to F if D has at least n vertices and min{d⁺(u)+d⁺(v),d⁻(u)+d⁻(v),d⁺(u)+d⁻(v)}≥2m−1 for every pair of vertices u,v∈V(D) with uv∉A(D). This is a common generalization of two results of Babu and Diwan, one on the existence of forests in graphs under a degree sum condition and the other on the existence of oriented forests in digraphs under a minimum degree condition.     

Weak cycle partition involving degree sum conditions

Let G be a graph of order n and k a positive integer. A set of subgraphs H={H₁,H₂,…,H_k} is called a k-degenerated cycle partition (abbreviated to k-DCP) of G if H₁,…,H_k are vertex disjoint subgraphs of G such that and for all i, 1≤i≤k, H_i is a cycle or K₁ or K₂. If, in addition, for all i, 1≤i≤k, H_i is a cycle or K₁, then H is called a k-weak cycle partition (abbreviated to k-WCP) of G. It has been shown by Enomoto and Li that if |G|=n≥k and if the degree sum of any pair of nonadjacent vertices is at least n−k+1, then G has a k-DCP, except G≅C₅ and k=2. We prove that if G is a graph of order n≥k+12 that has a k-DCP and if the degree sum of any pair of nonadjacent vertices is at least , then either G has a k-WCP or k=2 and G is a subgraph of K₂∪K_n−2∪{e}, where e is an edge connecting V(K₂) and V(K_n−2). By using this, we improve Enomoto and Li’s result for n≥max{k+12,10k−9}.     

Minimum vertex degree conditions for loose Hamilton cycles in 3-uniform hypergraphs

We investigate minimum vertex degree conditions for 3-uniform hypergraphs which ensure the existence of loose Hamilton cycles. A loose Hamilton cycle is a spanning cycle in which only consecutive edges intersect and these intersections consist of precisely one vertex.     

Edge disjoint Hamilton cycles in sparse random graphs of minimum degree at least k

Let G_n,m,k denote the space of simple graphs with n vertices, m edges, and minimum degree at least k, each graph G being equiprobable. Let G have property A_k, if G contains ⌊(k − 1)/2⌋ edge disjoint Hamilton cycles, and, if k is even, a further edge disjoint matching of size ⌊n/2⌋. We prove that, for k ≥ 3, there is a constant C_k such that if 2m ≥ C_kn then A_k occurs in G_n,m,k with probability tending to 1 as n → ∞. © 2000 John Wiley & Sons, Inc. J. Graph Theory 34: 42–59, 2000     

Spanning 2‐trails from degree sum conditions

Suppose G is a simple connected n‐vertex graph. Let σ₃(G) denote the minimum degree sum of three independent vertices in G (which is ∞ if G has no set of three independent vertices). A 2‐trail is a trail that uses every vertex at most twice. Spanning 2‐trails generalize hamilton paths and cycles. We prove three main results. First, if σ₃G)≥ n ‐ 1, then G has a spanning 2‐trail, unless G ? K_1,3. Second, if σ₃(G) ≥ n, then G has either a hamilton path or a closed spanning 2‐trail. Third, if G is 2‐edge‐connected and σ₃(G) ≥ n, then G has a closed spanning 2‐trail, unless G ? K_2,3 or K (the 6‐vertex graph obtained from K_2,3 by subdividing one edge). All three results are sharp. These results are related to the study of connected and 2‐edge‐connected factors, spanning k‐walks, even factors, and supereulerian graphs. In particular, a closed spanning 2‐trail may be regarded as a connected (and 2‐edge‐connected) even [2,4]‐factor. © 2004 Wiley Periodicals, Inc. J Graph Theory 45: 298–319, 2004     

CENTER CONDITIONS AND BIFURCATION OF LIMIT CYCLES FOR A CLASS OF FIFTH DEGREE SYSTEMS

The center conditions and bifurcation of limit cycles for a class of fifth degree systems are investigated. Two recursive formulas to compute singular quantities at infinity and at the origin are given. The first nine singular point quantities at infinity and first seven singular point quantities at the origin for the system are given in order to get center conditions and study bifurcation of limit cycles. Two fifth degree systems are constructed. One allows the appearance of eight limit cycles in the neighborhood of infinity,which is the first example that a polynomial differential system bifurcates eight limit cycles at infinity. The other perturbs six limit cycles at the origin.