It is known that Θ(log n) chords must be added to an n-cycle to produce a pancyclic graph; for vertex pancyclicity, where every vertex belongs to a cycle of every length, Θ(n) chords are required. A possibly ‘intermediate’ variation is the following: given k, 1 ≤ k ≤ n, how many chords must be added to ensure that there exist cycles of every possible length each of which passes exactly k chords...

We show that for each η > 0 every digraph G of sufficiently large order n is Hamiltonian if its outand indegree sequences d+1 ≤ · · · ≤ d + n and d−1 ≤ · · · ≤ d − n satisfy (i) d + i ≥ i+ ηn or d − n−i−ηn ≥ n− i and (ii) d − i ≥ i+ ηn or d+n−i−ηn ≥ n− i for all i < n/2. This gives an approximate solution to a problem of Nash-Williams [22] concerning a digraph analogue of Chvátal’s theorem. In ...

An edge-colored graph is \emph{rainbow }if no two edges of the have same color. $G^c$ called \emph{properly colored} if every adjacent receive distinct colors in $G^c$. A \emph{strongly edge-colored} a proper such that path length $3$ rainbow. We call an vertex pair-pancyclic} any vertices are contained rainbow cycle $\ell$ for each with $3 \leq \ell n$. In this paper, we show strongly order $n...

The balanced hypercube BHn is a variant of the hypercube Qn. Huang and Wu proved that BHn has better properties than Qn with the same number of links and processors. In particularly, they showed that there exists a cycle of length 2 l in BHn for all l, 2 6 l 6 2n. In this paper, we improve this result by showing that BHn is edge-pancyclic, which means that for arbitrary edge e, there exists a c...

A side skirt is a planar rooted tree T, T≠P2, where the root of T vertex degree at least two, and all other vertices except leaves are three. reduced Halin graph or skirted plane G=T∪P, skirt, P path connecting in order determined by embedding T. The structure graphs contains both symmetry asymmetry. For n≥2 Pn=v1v2v3⋯vn as length n−1, we call Cartesian product G Pn, n-generalized prism over G....

We pose and solve the existence of 2-factorizations of complete graphs and complete bipartite graphs that have the number of cycles per 2-factor varying, called pancomponented. Such 2-factorizations exist for all such graphs. The pancomponented problem requires a slight generalization of the methods used to solve pancyclic 2-factorization problem, by building 2-factors from cyclically generated...

a r t i c l e i n f o a b s t r a c t In a recently published paper [Z.-J. Xue, S.-Y. Liu, An optimal result on fault-tolerant cycle-embedding in alternating group graphs, Inform. the authors showed that for a set of faulty vertices F in an n-dimensional alternating group graph AG n , AG n − F remains pancyclic if |F | 2n − 6. However, the proof of the result is flawed. We will prove the theore...