On long cycles in a 2-connected bipartite graph

Letk be an integer withk 2. LetG = (A, B; E) be a 2-connected bipartite graph. Supposed(x) + d(y) k + 1 for every pair of non-adjacent verticesx andy. ThenG contains a cycle of length at leastmin(2a, 2k) wherea = min(|A|,|B|), unlessG is one of some known exceptions. We conjecture that if|A| = |B| andd(x) + d(y) k + 1 for every pair of non-adjacent verticesx andy withx A andy B, thenG contains a cycle of length at leastmin(2a, 2k).     

Disjoint triangles and quadrilaterals in a graph

Let n, s and t be three integers with s ≥ 1, t ≥ 0 and n = 3s + 4t. Let G be a graph of order n such that the minimum degree of G is at least (n + s)/2. Then G contains a 2-factor with s + t components such that s of them are triangles and t of them are quadrilaterals.      

Disjoint cycles in digraphs

We show that, for each natural numberk, these exists a (smallest) natural numberf(k) such that any digraph of minimum outdegree at leastf(k) containsk disjoint cycles. We conjecture thatf(k)=2k−1 and verify this fork=2 and we show that, for eachk≧3, the determination off(k) is a finite problem. Dedicated to Paul Erdős on his seventieth birthday     

Disjoint triangles and quadrilaterals in a graph

Let G be a simple graph of order n and s and k be two positive integers. Brandt et al. obtained the following result: If s?k, n?3s+4(k-s) and σ₂(G)?n+s, then G contains k disjoint cycles C₁,…,C_k satisfying |C_i|=3 for 1?i?s and |C_i|?4 for s<i?k. In the above result, the length of C_i is not specified for s<i?k. We get a result specifying the length of C_i for each s<i?k if n?3s+4(k-s)+3.     

Disjoint cycles in star-free graphs

A graph is claw-free if it does not contain K_1.3 as an induced subgraph. It is K_1.r-free if it does not contain K_1.r as an induced subgraph. We show that if a graph is K_1.r-free (r ≥ 4), only p + 2r − 1 edges are needed to insure that G has two disjoint cycles. As an easy consequence we get a well-known result of Pósa's. © 1996 John Wiley & Sons, Inc.     

Disjoint chorded cycles in graphs

We propose the following conjecture to generalize results of Pósa and of Corrádi and Hajnal. Let r,s be nonnegative integers and let G be a graph with |V(G)|≥3r+4s and minimal degree δ(G)≥2r+3s. Then G contains a collection of r+s vertex disjoint cycles, s of them with a chord. We prove the conjecture for r=0,s=2 and for s=1. The corresponding extremal problem, to find the minimum number of edges in a graph on n vertices ensuring the existence of two vertex disjoint chorded cycles, is also settled.     

Disjoint hamiltonian cycles in bipartite graphs

Let G=(X,Y) be a bipartite graph and define . Moon and Moser [J. Moon, L. Moser, On Hamiltonian bipartite graphs, Israel J. Math. 1 (1963) 163-165. MR 28 # 4540] showed that if G is a bipartite graph on 2n vertices such that , then G is hamiltonian, sharpening a classical result of Ore [O. Ore, A note on Hamilton circuits, Amer. Math. Monthly 67 (1960) 55] for bipartite graphs. Here we prove that if G is a bipartite graph on 2n vertices such that , then G contains k edge-disjoint hamiltonian cycles. This extends the result of Moon and Moser and a result of R. Faudree et al. [R. Faudree, C. Rousseau, R. Schelp, Edge-disjoint Hamiltonian cycles, Graph Theory Appl. Algorithms Comput. Sci. (1984) 231-249].     

Disjoint homotopic paths and trees in a planar graph

In this paper we describe a polynomial-time algorithm for the following problem:given: a planar graphG embedded in ℝ², a subset {I ₁, …,I _p} of the faces ofG, and pathsC ₁, …,C _k inG, with endpoints on the boundary ofI ₁ ∪ … ∪I _p; find: pairwise disjoint simple pathsP ₁, …,P _k inG so that, for eachi=1, …,k, P _i is homotopic toC _i in the space ℝ²\(I ₁ ∪ … ∪I _p). Moreover, we prove a theorem characterizing the existence of a solution to this problem. Finally, we extend the algorithm to disjoint homotopic trees. As a corollary we derive that, for each fixedp, there exists a polynormial-time algorithm for the problem:given: a planar graphG embedded in ℝ² and pairwise disjoint setsW ₁, …,W _k of vertices, which can be covered by the boundaries of at mostp faces ofG;find: pairwise vertex-disjoint subtreesT ₁, …,T _k ofG whereT _i (i=1, …, k).     

Vertex-disjoint chorded cycles in a graph

In this paper we prove: Let k≥1 be an integer and G be graph with at least 4k vertices and minimum degree at least ⌊7k/2⌋. Then G contains k vertex-disjoint cycles such that each of them has at least two chords in G.     

Two vertex-disjoint cycles in a graph

LetG be a graph of ordern 6 with minimum degree at least (n + 1)/2. Then, for any two integerss andt withs 3,t 3 ands + t n, G contains two vertex-disjoint cycles of lengthss andt, respectively, unless thatn, s andt are odd andG is isomorphic toK _{(n–1)/2,(n–1)/2} + K₁. We also show that ifG is a graph of ordern 8 withn even and minimum degree at leastn/2, thenG contains two vertex-disjoint cycles with any given even lengths provided that the sum of the two lengths is at mostn.     

Covering a graph with cycles

Let k and n be two integers such that k ≥ 0 and n ≥ 3(k + 1). Let G be a graph of order n with minimum degree at least ?(n + k)/2?. Then G contains k + 1 independent cycles covering all the vertices of G such that k of them are triangles. © 1995, John Wiley & Sons, Inc.     

Disjoint Hamiltonian cycles in fan 2k-type graphs

It is conjectured that a 2(k + 1)-connected graph of order p contains k + 1 pairwise disjoint Hamiltonian cycles if no two of its vertices that have degree less than 1/2 + 2k are distance two apart. This is proved in detail for k = 1. Similar arguments establish the conjecture for k = 2. © 1993 John Wiley & Sons, Inc.     

Fan-type condition on disjoint cycles in a graph

Let $k$ be a positive integer. In this paper, we prove that for a graph $G$ with at least $4k$ vertices, if max$\{d\left(x\right),d\left(y\right)\}\ge 2k$ for any pair of nonadjacent vertices $\{x,y\}?V\left(G\right)$, then $G$ contains $k$ disjoint cycles. This generalizes the results given by Corrá di and Hajnal (1963), Enomoto (1998), and Wang (1999).     

Independent cycles with limited size in a graph

Letk and s be two positive integers with s≥3. LetG be a graph of ordern ≥sk. Writen =qk + r, 0 ≤r ≤k - 1. Suppose thatG has minimum degree at least (s - l)k. Then G containsk independent cyclesC ₁,C ₂,...,C _k such thats ≤l(C _i ) ≤q for 1 ≤i ≤r arnds ≤l(C _i ) ≤q + 1 fork -r <i ≤k, where l(C_i) denotes the length ofC _i .     

Disjoint cycles with different length in 4-arc-dominated digraphs

A d$d$-arc-dominated digraph is a digraph D$D$ of minimum out-degree d$d$ such that for every arc (x,y)$(x,y)$ of D$D$, there exists a vertex u$u$ of D$D$ of out-degree d$d$ such that (u,x)$(u,x)$ and (u,y)$(u,y)$ are arcs of D$D$. Henning and Yeo [Vertex disjoint cycles of different length in digraphs, SIAM J. Discrete Math. 26 (2012) 687–694] conjectured that a digraph with minimum out-degree at least four contains two vertex-disjoint cycles of different length. In this paper, we verify this conjecture for 4-arc-dominated digraphs.