A graph G on n vertices is said to be (k, m)-pancyclic if each k-set S ⊆ V (G) is contained in cycles of each of the following lengths: m,m + 1, · · · , n. This property generalizes vertex pancyclicity. There are ten pairs of forbidden subgraphs which guarantee that a 2-connected graph is (k,m)-pancyclic for some integer m ≤ n. We give the best (smallest) possible value for m in each of these t...

As an enhancement on the hypercube Qn, the augmented cube AQn, prosed by Choudum and Sunitha [S.A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2) (2002) 71–84], not only retains some favorable properties of Qn but also possesses some embedding properties that Qn does not. For example, AQn is pancyclic, that is, AQn contains cycles of arbitrary length for n 2. This paper shows that AQn re...

We deal with different algorithmic questions regarding properly arc-colored s-t paths, trails and circuits in arc-colored digraphs. Given an arc-colored digraph D with c ≥ 2 colors, we show that the problem of maximizing the number of arc disjoint properly arc-colored s-t trails can be solved in polynomial time. Surprisingly, we prove that the determination of one properly arc-colored s-t path ...

A hypergraph $\mathcal H$ is super-pancyclic if for each $A \subseteq V(\mathcal H)$ with $|A| \geqslant 3$, contains a Berge cycle base vertex set $A$. We present two natural necessary conditions to be super-pancyclic, and show that in several classes of hypergraphs these are also sufficient. In particular, they sufficient every $ \delta(\mathcal H)\geqslant \max\{|V(\mathcal H)|, \frac{|E(\ma...

In an edge-colored graph (G,c), let dc(v) denote the number of colors on edges incident with a vertex v G and δc(G) minimum value over all vertices v∈V(G). A cycle (G,c) is called proper if any two adjacent have distinct colors. An n≥3 properly vertex-pancyclic each contained in length ℓ for every 3≤ℓ≤n. Fujita Magnant conjectured that complete δc(G)≥n+12 vertex-pancyclic. Chen, Huang Yuan part...