Heavy cycles passing through some specified vertices in weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex υ is called the weighted degree of υ. The weight of a cycle is defined as the sum of the weights of its edges. In this paper, we prove that: (1) if G is a 2‐connected weighted graph such that the minimum weighted degree of G is at least d, then for every given vertices x and y, either G contains a cycle of weight at least 2d passing through both of x and y or every heaviest cycle in G is a hamiltonian cycle, and (2) if G is a 2‐connected weighted graph such that the weighted degree sum of every pair of nonadjacent vertices is at least s, then for every vertex y, G contains either a cycle of weight at least s passing through y or a hamiltonian cycle. AMS classification: 05C45 05C38 05C35. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Long cycles passing through vertices in k‐connected graphs

In this article, we consider the following problem proposed by Locke and Zhang in 1991: Let G be a k‐connected graph with minimum degree d and X a set of m vertices on a cycle of G. For which values of m and k, with , must G have a cycle of length at least passing through X? Fujisawa and Yamashita solved this problem for the case and in 2008. We provide an affirmative answer to this problem for the case of and .     

Covering a graph with cycles passing through given edges

We propose a conjecture: for each integer k ≥ 2, there exists N(k) such that if G is a graph of order n ≥ N(k) and d(x) + d(y) ≥ n + 2k - 2 for each pair of non-adjacent vertices x and y of G, then for any k independent edges e₁, …, e_k of G, there exist k vertex-disjoint cycles C₁, …, C_k in G such that e_i ∈ E(C_i) for all i ∈ {1, …, k} and V(C₁ ∪ ···∪ C_k) = V(G). If this conjecture is true, the condition on the degrees of G is sharp. We prove this conjecture for the case k = 2 in the paper. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 105–109, 1997     

Hamilton decompositions of complete multipartite graphs with any 2-factor leave

For m ≥ 1 and p ≥ 2, given a set of integers s₁,…,s_q with for and , necessary and sufficient conditions are found for the existence of a hamilton decomposition of the complete p-partite graph , where U is a 2-factor of consisting of q cycles, the jth cycle having length s_j. This result is then used to completely solve the problem when p = 3, removing the condition that . © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 208–214, 2003     

Cycles through three vertices in 2-connected graphs

Let Gbe a 2-connected graph with minimum degree d and let {x, y, z} be a set of three vertices contained on some cycle ofG. ThenG ishamiltonian or {x, y, z} is contained on a cycle of length at least 2d inG.     

On cycles through specified vertices

For a set X of vertices of a graph fulfilling local connectedness conditions, the existence of a cycle containing X is proved.     

Decomposition of cubic graphs with a 2-factor consisting of three cycles

The 3-Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree, a 2-regular subgraph and a matching. We prove that this conjecture is true for connected cubic graphs with a 2-factor consisting of three cycles.     

Counterexamples to a conjecture of mader about cycles through specified vertices inn-edge-connected graphs

For each oddn≥3, we constructn-edge-connected graphsG with the following property: There are two verticesu andv inG such that for every cycleC inG passign throughu andv the graphG-E(C) is not (n-2)-edge-connected. HereE(C) denotes the set of edges ofC, and a cycle is allowed to pass through a vertex more than once.     

Edge disjoint cycles through specified vertices

We give a sufficient condition for a simple graph G to have k pairwise edge‐disjoint cycles, each of which contains a prescribed set W of vertices. The condition is that the induced subgraph G[W] be 2k‐connected, and that for any two vertices at distance two in G[W], at least one of the two has degree at least |V(G)|/2 + 2(k ? 1) in G. This is a common generalization of special cases previously obtained by Bollobás/Brightwell (where k = 1) and Li (where W = V(G)). A key lemma is of independent interest. Let G be the complement of a bipartite graph with partite sets X, Y. If G is 2k connected, then G contains k Hamilton cycles that are pairwise edge‐disjoint except for edges in G[Y]. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Cycles passing through k + 1 vertices in k‐connected graphs

In this article, we prove the following theorem. Let k ≥ 3 be an integer, G be a k‐connected graph with minimum degree d and X be a set of k + 1 vertices on a cycle. Then G has a cycle of length at least min {2d,|V(G)|} passing through X. This result gives the positive answer to the Question posed by Locke [8]. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:179–190, 2008     

The Wiener index of unicyclic graphs given number of pendant vertices or cut vertices

In this paper, we consider the conjugate-Toeplitz (CT) and conjugate-Hankel (CH) matrices. It is proved that the inverse of these special matrices can be expressed as the sum of products of lower and upper triangular matrices. Firstly, we get access to the explicit inverse of conjugate-Toeplitz matrix. Secondly, the decomposition of the inverse is obtained. Similarly, the formulae and the decomposition on inverse of conjugate-Hankel are provided. Thirdly, the stability of the inverse formulae of CT and CH matrices are discussed. Finally, examples are provided to verify the feasibility of the algorithms provided in this paper.     

Embedding cycles of given length in oriented graphs

Kelly, Kühn and Osthus conjectured that for any ?≥4$?\ge 4$ and the smallest number k≥3$k\ge 3$ that does not divide ?$?$, any large enough oriented graph G$G$ with δ⁺(G),δ⁻(G)≥⌊|V(G)|/k⌋+1${\delta }^{+}\left(G\right),{\delta }^{-}\left(G\right)\ge \lfloor \left|V\left(G\right)\right|/k\rfloor +1$ contains a directed cycle of length ?$?$. We prove this conjecture asymptotically for the case when ?$?$ is large enough compared to k$k$ and k≥7$k\ge 7$. The case when k≤6$k\le 6$ was already settled asymptotically by Kelly, Kühn and Osthus.     

Finding any given 2‐factor in sparse pseudorandom graphs efficiently

Given an $n$‐vertex pseudorandom graph $G$ and an $n$‐vertex graph $H$ with maximum degree at most two, we wish to find a copy of $H$ in $G$, that is, an embedding $\phi :V\left(H\right)\to V\left(G\right)$ so that $\phi \left(u\right)\phi \left(v\right)\in E\left(G\right)$ for all $uv\in E\left(H\right)$. Particular instances of this problem include finding a triangle‐factor and finding a Hamilton cycle in $G$. Here, we provide a deterministic polynomial time algorithm that finds a given $H$ in any suitably pseudorandom graph $G$. The pseudorandom graphs we consider are $\left(p,\lambda \right)$‐bijumbled graphs of minimum degree which is a constant proportion of the average degree, that is, $\mathrm{\Omega }\left(pn\right)$. A $\left(p,\lambda \right)$‐bijumbled graph is characterised through the discrepancy property: $|e\left(A,B\right)?p|A||B||<\lambda \sqrt{|A||B|}$ for any two sets of vertices $A$ and $B$. Our condition $\lambda =O(p^{2}n/\log n)$ on bijumbledness is within a log factor from being tight and provides a positive answer to a recent question of Nenadov. We combine novel variants of the absorption‐reservoir method, a powerful tool from extremal graph theory and random graphs. Our approach builds on our previous work, incorporating the work of Nenadov, together with additional ideas and simplifications.     

Cycles through 4 vertices in 3-connected graphs

S.C. Locke proposed a question: If G is a 3-connected graph with minimum degree d and X is a set of 4 vertices on a cycle in G, must G have a cycle through X with length at least min{2d,|V(G)|}? In this paper, we answer this question.     

Hamiltonian cycles and dominating cycles passing through a linear forest

Let G be an (m+2)-graph on n vertices, and F be a linear forest in G with |E(F)|=m and ω₁(F)=s, where ω₁(F) is the number of components of order one in F. We denote by σ₃(G) the minimum value of the degree sum of three vertices which are pairwise non-adjacent. In this paper, we give several σ₃ conditions for a dominating cycle or a hamiltonian cycle passing through a linear forest. We first prove that if σ₃(G)≥n+2m+2+max{s−3,0}, then every longest cycle passing through F is dominating. Using this result, we prove that if σ₃(G)≥n+κ(G)+2m−1 then G contains a hamiltonian cycle passing through F. As a corollary, we obtain a result that if G is a 3-connected graph and σ₃(G)≥n+κ(G)+2, then G is hamiltonian-connected.