The technique of counting cliques in networks is a natural problem. In this paper, we develop certain results on counting of triangles for the total graph of the Mycielski graph or central graph of star as well as completegraph families. Moreover, we discuss the upper bounds for the number of triangles in the Mycielski and other well known transformations of graphs. Finally, it is shown that th...

Let G = (m,n) be an undirected graph with m edges and n vertices. For a random walk on G it is known that the time to cover all its edges is bounded by O(mn) [2]. In a later work the bound O(m2) [3] is proved, which holds even for graphs with weighted edges. Here, we briefly discuss these results along with their proofs.

In this paper we de1ne the vertex-cover polynomial (G; ) for a graph G. The coe2cient of r in this polynomial is the number of vertex covers V ′ of G with |V ′|= r. We develop a method to calculate (G; ). Motivated by a problem in biological systematics, we also consider the mappings f from {1; 2; : : : ; m} into the vertex set V (G) of a graph G, subject to f−1(x) ∪ f−1(y) = ∅ for every edge x...

A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An L-cycle cover is a cycle cover in which the length of every cycle is in the set L. A special case of L-cycle covers are k-cycle covers for k ∈ N, where the length of each cycle must be at least k. The weight of a cycle cover of an edge-weighted graph is the sum of the weights of its edges. We com...

A biclique B of a simple graph G is the edge-set of a complete bipartite subgraph of G. A biclique cover of G is a collection of bicliques covering the edge-set of G. Given a graph G, we will study the following problem: find the minimum number of bicliques which cover the edge-set of G. This problem will be called the minimum biclique cover problem (MBC). First, we will define the families of ...

1. • Let T = (V, E) be a tree and let D =< d1, . . . , dk > denote the set of demands. We reduce the prize-collecting steiner forest problem on trees to the minimum weighted vertex cover problem. We make a bipartite graph H = (V ′, E ′) where for each edge e ∈ E we put a vertex pe in V ′ and also for every di ∈ D we put a vertex qi in V ′. We put an edge between pe and qi iff the edge e is on t...

We consider the one-variable characteristic polynomial p(G; ) in two settings. When G is a rooted digraph, we show that this polynomial essentially counts the number of sinks in G. When G is a rooted graph, we give combinatorial interpretations of several coe/cients and the degree of p(G; ). In particular, |p(G; 0)| is the number of acyclic orientations of G, while the degree of p(G; ) gives th...

We consider the problem of maintaining a large matching or a small vertex cover in a dynamically changing graph. Each update to the graph is either an edge deletion or an edge insertion. We give the first data structure that simultaneously achieves a constant approximation factor and handles a sequence of k updates in k · polylog(n) time. Previous data structures require a polynomial amount of ...

Consider a tree T and a forest F . We discuss the following problems. Problem FVC: cover the vertices of T by a minimum number of copies of trees of F , such that every vertex of T is covered exactly once. Problem FEC: cover the edges of T with a minimum number of copies of trees of F , such that every edge of T is covered exactly once. F contains a one vertex (one edge) tree to always have a s...