in this paper, it was shown that , where  and , and , where  is not prime and , are od-characterizable.

The focus of research in this paper is to investigate whether a random network whose degree sequence of the vertices is the same as the degree sequence of the vertices in a real-world network would exhibit values for other analysis metrics similar to those of the real-world network. We use the well-known Configuration Model to generate a random network on the basis of the degree sequence of the...

Given positive integers k m n, a graphG of order n is ðk;mÞ-pancyclic if for any set of k vertices of G and any integer r with m r n, there is a cycle of length r containing the k vertices. Minimum degree conditions and minimum sum of degree conditions of nonadjacent vertices that imply a graph is ðk;mÞ-pancylic are proved. If the additional property that the k vertices must appear on the cycle...

How small can a graph be that contains as subgraphs all trees on n vertices with maximum degree d? In this paper, this question is answered by constructing such universal graphs that have n vertices and bounded degree (depending only on d). Universal graphs with n vertices and O(n log n) edges are also constructed that contain all bounded-degree planar graphs on n vertices as subgraphs. In gene...

Let $G$ be a finite group and $pi(G)$ be the set of all the prime‎ ‎divisors of $|G|$‎. ‎The prime graph of $G$ is a simple graph‎ ‎$Gamma(G)$ whose vertex set is $pi(G)$ and two distinct vertices‎ ‎$p$ and $q$ are joined by an edge if and only if $G$ has an‎ ‎element of order $pq$‎, ‎and in this case we will write $psim q$‎. ‎The degree of $p$ is the number of vertices adjacent to $p$ and is‎ ...

By a theorem of Mader [5], highly connected subgraphs can be forced in finite graphs by assuming a high minimum degree. Solving a problem of Diestel [2], we extend this result to infinite graphs. Here, it is necessary to require not only high degree for the vertices but also high vertex-degree (or multiplicity) for the ends of the graph, i.e. a large number of disjoint rays in each end. We give...

Sumner’s universal tournament conjecture states that any tournament on 2n−2 vertices contains a copy of any directed tree on n vertices. We prove an asymptotic version of this conjecture, namely that any tournament on (2 + o(1))n vertices contains a copy of any directed tree on n vertices. In addition, we prove an asymptotically best possible result for trees of bounded degree, namely that for ...

Let G be an edge-colored graph and v a vertex of G. The color degree of v is the number of colors appearing on the edges incident to v. A rainbow triangle in G is one in which all edges have distinct colors. In this paper, we first prove that an edge-colored graph on n vertices contains a rainbow triangle if the color degree sum of any two adjacent vertices is at least n+ 1. Afterwards, we char...

For a planar graph on n vertices we determine the maximum values for the following: 1) the sum of the m largest vertex degrees. 2) the number of vertices of degree at least k. 3) the sum of the degrees of vertices with degree at least k.

Böttcher, Schacht and Taraz [6] gave a condition on the minimum degree of a graph G on n vertices that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth o(n), thereby proving a conjecture of Bollobás and Komlós [15]. We strengthen this result in the case when H is bipartite. Indeed, we give an essentially best-possible condition on the degree sequence...