The edge clique cover number ecc ( G ) of a graph is the size smallest collection complete subgraphs whose union covers all edges G. Chen, Jacobson, Kézdy, Lehel, Scheinerman, and Wang conjectured in 2000 that if claw-free, then bounded above by its order (denoted n). Recently, Javadi Hajebi verified this conjecture for claw-free graphs with an independence at least three. We study two, which a...

Let G n $$ {G}_n be a random geometric graph, and then for q , p ∈ [ 0 1 ) q,p\in \left[0,1\right) we construct ( \left(q,p\right) -perturbed noisy graph {G}_n^{q,p} where each existing edge in is removed with probability while non-existent inserted . We give asymptotically tight bounds on the clique number ω \omega \left({G}_n^{q,p}\right) several regimes of parameter.

In this paper we compare and illustrate the algorithmic use of graphs of bounded treewidth and graphs of bounded clique-width. For this purpose we give polynomial time algorithms for computing the four basic graph parameters independence number, clique number, chromatic number, and clique covering number on a given tree structure of graphs of bounded tree-width and graphs of bounded clique-widt...

Several new tools are presented for determining the number of cliques needed to (edge-)partition a graph . For a graph on n vertices, the clique partition number can grow cn z times as fast as the clique covering number, where c is at least 1/64. If in a clique on n vertices, the edges between en° vertices are deleted, Z--a < 1, then the number of cliques needed to partition what is left is asy...

A clique-transversal of a graph G is a subset of vertices that meets all the cliques of G. A clique-independent set is a collection of pairwise vertex-disjoint cliques. The clique-transversal number and clique-independence number of G are the sizes of a minimum clique-transversal and a maximum clique-independent set of G, respectively. A graph G is clique-perfect if the sizes of a minimum cliqu...

We consider the problem of determining the size of a maximum clique in a graph, also known as the clique number. Given any method that computes an upper bound on the clique number of a graph, we present a sequential elimination algorithm which is guaranteed to improve upon that upper bound. Computational experiments on DIMACS instances show that, on average, this algorithm can reduce the gap be...

Chudnovsky and Seymour proved that every connected claw-free graph that contains a stable set of size 3 has chromatic number at most twice its clique number. We improve this for small clique size, showing that every claw-free graph with clique number at most 3 is 4-choosable and every claw-free graph with clique number at most 4 is 7-choosable. These bounds are tight.

A clique-transversal of a graph G is a subset of vertices that meets all the cliques of G. A clique-independent set is a collection of pairwise vertex-disjoint cliques. The clique-transversal number and clique-independence number of G are the sizes of a minimum clique-transversal and a maximum clique-independent set of G, respectively. A graph G is clique-perfect if these two numbers are equal ...

For a graph $G = (V, E)$, a partition $pi = {V_1,$ $V_2,$ $ldots,$ $V_k}$ of the vertex set $V$ is an textit{upper domatic partition} if $V_i$ dominates $V_j$ or $V_j$ dominates $V_i$ or both for every $V_i, V_j in pi$, whenever $i neq j$. The textit{upper domatic number} $D(G)$ is the maximum order of an upper domatic partition. We study the properties of upper domatic number and propose an up...