Vertex Covers by Edge Disjoint Cliques

Vertex covers by edge disjoint

Let H be a simple graph having no isolated vertices. An (H; k)-vertex-cover of a simple graph G = (V;E) is a collection H1; : : : ;Hr of subgraphs of G satisfying 1. Hi = H; for all i = 1; : : : ; r; 2. [i=1V (Hi) = V , 3. E(Hi) \ E(Hj) = ;; for all i 6= j; and 4. each v 2 V is in at most k of the Hi. We consider the existence of such vertex covers when H is a complete graph, Kt; t 3, in the co...

Turán Numbers of Vertex-disjoint Cliques in r-Partite Graphs

graph is called a complete r-partite graph if its vertex set can be partitioned into r independent sets V1, . . . , Vr such that for any i = 1, 2, . . . , r every vertex in Vi is adjacent to all other vertices in Vj , j 6= i. We denote a complete r-partite graph with part sizes |Vi| = ni by Kn1,...,nr . For a graph G and a positive integer k we use kG to denote k vertex-disjoint copies of G. Gi...

Turán numbers of vertex-disjoint cliques in -partite graphs

Edge dominating sets and vertex covers

Bipartite graphs with equal edge domination number and maximum matching cardinality are characterized. These two parameters are used to develop bounds on the vertex cover and total vertex cover numbers of graphs and a resulting chain of vertex covering, edge domination, and matching parameters is explored. In addition, the total vertex cover number is compared to the total domination number of ...

Edge-Disjoint Cliques in Graphs with High Minimum Degree

For a graph G and a fixed integer k ≥ 3, let νk(G) denote the maximum number of pairwise edge-disjoint copies of Kk in G. For a constant c, let η(k, c) be the infimum over all constants γ such that any graph G of order n and minimum degree at least cn has νk(G) ≥ γn2(1 − on(1)). By Turán’s Theorem, η(k, c) = 0 if c ≤ 1− 1/(k− 1) and by Wilson’s Theorem, η(k, c) → 1/(k2 − k) as c → 1. We prove t...