As an enhancement on the hypercube Qn, the augmented cube AQn, proposed by Choudum and Sunitha [S.A. Choudum, V. Sunitha, Augmented cubes, Networks, 40(2) (2002), 71–84], not only retains some of the favorable properties of Qn but also possesses some embedding properties that Qn does not. For example, AQn contains cycles of all lengths from 3 to 2, but Qn contains only even cycles. In this pape...

Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar. Preface Graph theory as a very popular area of discrete mathematics has rapidly been developed over the last couple of decades. Numerous theoretical results and countless applications to practical problems have been discovered. The concepts of k-ordered graphs and out-arc pancyclicity are two recent topics i...

Let $T$ be a non-trivial tournament. An arc is emph{$t$-pancyclic} in $T$, if it is contained in a cycle of length $ell$ for every $tleq ell leq |V(T)|$. Let $p^t(T)$ denote the number of $t$-pancyclic arcs in $T$ and $h^t(T)$ the maximum number of $t$-pancyclic arcs contained in the same Hamiltonian cycle of $T$. Moon ({em J. Combin. Inform. System Sci.}, {bf 19} (1994), 207-214) showed that $...

A graph G on n vertices is Hamiltonian if it contains a spanning cycle, and pancyclic if it contains cycles of all lengths from 3 to n. In 1984, Fan presented a degree condition involving every pair of vertices at distance two for a 2-connected graph to be Hamiltonian. Motivated by Fan’s result, we say that an induced subgraph H of G is f1-heavy if for every pair of vertices u, v ∈ V (H), dH(u,...

The toughness $t(G)$ of a connected graph $G$ is defined as $t(G)=\min\{\frac{|S|}{c(G-S)}\}$, in which the minimum taken over all proper subsets $S\subset V(G)$ such that $c(G-S)>1$, where $c(G-S)$ denotes number components $G-S$. Let $\lambda$ denote second largest absolute eigenvalue adjacency matrix graph. For any $d$-regular $G$, it has been shown by Alon $t(G)>\frac{1}{3}(\frac{d^2}{d\lam...

A digraph of order at least k is termed k-traceable if each of its subdigraphs of order k is traceable. It turns out that several properties of tournaments—i.e., the 2-traceable oriented graphs—extend to k-traceable oriented graphs for small values of k. For instance, the authors together with O. Oellermann have recently shown that for k = 2, 3, 4, 5, 6, all ktraceable oriented graphs are trace...

Let G x H denote the Kronecker product of graphs G and H. Principal results are as follows: (a) If m is even and n 0 (mod 4), then one component of P,.+l x P,+1, and each component of each of CA x Pn+l, Pm+l x (7, and Cm x C, are edge decomposable into cycles of uniform length rs, where r and s are suitable divisors of m and n, respectively, (b) if m and n are both even, then each component of ...

In this paper, we investigate a problem on embedding paths into recursive circulant G(2, 4) with faulty elements (vertices and/or edges) and show that each pair of vertices in recursive circulant G(2, 4), m ≥ 3, are joined by a fault-free path of every length from m + 1 to |V (G(2, 4)\F )| − 1 inclusive for any fault set F with |F | ≤ m− 3. The bound m−3 on the number of acceptable faulty eleme...

For any natural number k, a graph G is said to be pancyclic mod k if it contains a cycle of every length modulo k. In this paper, we show that every K1,4-free graph G with minimum degree δ(G) ≥ k+ 3 is pancyclic mod k and every claw-free graph G with δ(G) ≥ k+ 1 is pancyclic mod k, which confirms Thomassen’s conjecture [8] for claw-free graphs.

The (n, k)-star graph is a generalized version of the n-star graph, which belongs to the class of Cayley graphs, and has been recognized as an attractive alternative to an n-cube for building massively parallel computers. Recently, Chen et al. showed that an (n, k)-star graph is 6-weak-vertex-pancyclic for k < n–1, that is, each vertex of an (n, k)-star graph is contained in a cycle of length r...