A clique-coloring of a given graph G is a coloring of the vertices of G such that no maximal clique of size at least two is monocolored. The clique-chromatic number of G is the least number of colors for which G admits a clique-coloring. It has been proved that every planar graph is 3-clique colorable and every claw-free planar graph, different from an odd cycle, is 2-clique colorable. In this ...

The clique chromatic number of a graph is the smallest colors in vertex coloring so that no maximal monochromatic. In 2016 McDiarmid, Mitsche and Prałat noted around p ≈ n − 1 / 2 $$ p\approx {n}^{-1/2} random G , {G}_{n,p} changes by Ω ( ) {n}^{\Omega (1)} when we increase edge-probability o {n}^{o(1)} but left details this surprising “jump” phenomenon as an open problem. We settle problem, is...

A graph G is k-clique replete if G has clique number k and every elementary homomorphism of G has clique number greater than k . Results on the order of k-clique replete graphs are presented, and bounds for the minimum degree and the maximum degree of such graphs are discussed.

For a k-graph F?[n]k, the clique number of F is defined to be maximum size subset Q [n] with Qk?F. In present paper, we determine edges in on matching at most s and least q for n?8k2s q?(s+1)k?l, n?(s+1)k+s/(3k)?l. Two special cases that q=(s+1)k?2 k=2 are solved completely.

The packing chromatic number χρ(G) of a graphG is the smallest integer k such that the vertex set of G can be partitioned into sets Vi, i ∈ [k], where each Vi is an i-packing. In this paper, we investigate for a given triple (a, b, c) of positive integers whether there exists a graph G such that ω(G) = a, χ(G) = b, and χρ(G) = c. If so, we say that (a, b, c) is realizable. It is proved that b =...

The linear vertex-arboricity of a graph G is defined to the minimum number of subsets into which the vertex-set G can be partitioned so that every subset induces a linear forest. In this paper, we give the upper and lower bounds for sum and product of linear vertex-arboricity with independence number and with clique cover number respectively. All of these bounds are sharp.

Let Q(n, c) denote the minimum clique number over graphs with n vertices and chromatic number c. We investigate the asymptotics of Q(n, c) when n/c is held constant. We show that when n/c is an integer α, Q(n, c) has the same growth order as the inverse function of the Ramsey number R(α+ 1, t) (as a function of t). Furthermore, we show that if certain asymptotic properties of the Ramsey numbers...

Computing the clique number and chromatic number of a general graph are well-known to be NP-Hard problems. Codenotti et al. (Bruno Codenotti, Ivan Gerace, and Sebastiano Vigna. Hardness results and spectral techniques for combinatorial problems on circulant graphs. Linear Algebra Appl., 285(1-3): 123–142, 1998) showed that computing the clique number and chromatic number are still NP-Hard probl...

Let G be a finite abelian group of order n. For any subset B of G with B = −B, the Cayley graph GB is a graph on vertex set G in which ij is an edge if and only if i − j ∈ B. It was shown by Ben Green [3] that when G is a vector space over a finite field Z/pZ, then there is a Cayley graph containing neither a complete subgraph nor an independent set of size more than c logn log logn, where c > ...