the chromatic number of a graph g, denoted by χ(g), is the minimum number of colors such that g can be colored with these colors in such a way that no two adjacent vertices have the same color. a clique in a graph is a set of mutually adjacent vertices. the maximum size of a clique in a graph g is called the clique number of g. the turán graph tn(k) is a complete k-partite graph whose partition...

we consider a dynamic domination problem for graphs in which an infinitesequence of attacks occur at vertices with guards and the guard at theattacked vertex is required to vacate the vertex by moving to a neighboringvertex with no guard. other guards are allowed to move at the same time, andbefore and after each attack and the resulting guard movements, the verticescontaining guards form a dom...

let $g$ be a simple graph. an independent set is a set ofpairwise non-adjacent vertices. the number of vertices in a maximum independent set of $g$ isdenoted by $alpha(g)$. in this paper, we characterize graphs $g$ with $n$ vertices and with maximumnumber of maximum independent sets provided that $alpha(g)leq 2$ or $alpha(g)geq n-3$.

The independence number α ( G ) $\alpha (G)$ and the dissociation diss $\text{diss}(G)$ of a graph $G$ are largest orders induced subgraphs maximum degree at most 0 1, respectively. We consider possible improvements obvious inequality 2 ≥ $2\alpha (G)\ge \text{diss}(G)$ . For connected cubic graphs distinct from K 4 ${K}_{4}$ , we show 5 3 $5\alpha 3\,\text{diss}(G)$ describe rich interesting s...

The analogue of Hadwiger’s conjecture for the immersion order states that every graph G contains Kχ(G) as an immersion. If true, this would imply with n vertices and independence number α K⌈nα⌉ best currently known bound is due to Gauthier, Le Wollan, who recently proved a clique on ⌈χ(G)−43.54⌉ vertices. Their result implies n-vertex ⌈n3.54α−1.13⌉ We improve all α≥3, by showing α≥3 ⌊n2.25α−f(α...