Let G = (V,E) be a (finite, simple, undirected) graph with vertex set V and edge set E. A clique in G is a set of vertices, each of which is adjacent to every other vertex in this set. The maximum clique problem is to find a largest such set in a given graph; this is NP-hard [GJ90]. We denote the size of a maximum clique by ω. A colouring of G is an assignment of vertices to colours such that n...

A clique-colouring of a graph G is a colouring of the vertices of G so that no maximal clique of size at least two is monochromatic. The clique-hypergraph, H(G), of a graph G has V (G) as its set of vertices and the maximal cliques of G as its hyperedges. A vertex-colouring of H(G) is a clique-colouring of G. Determining the clique-chromatic number, the least number of colours for which a graph...

Bang-Jensen, J., P. Hell and G. MacGillivray, On the complexity of colouring by superdigraphs of bipartite graphs, Discrete Mathematics 109 (1992) 27-44. Let H be a directed graph whose vertices are called colours. An H-colouring of a digraph G is an assignment of these colours to the vertices of G so that if g is adjacent to g’ in G theq colour(g) is adjacent to colour(g’) in H (i.e., a homomo...

Eichhornia crassipes (Mart.) Solms is a free-floating aquatic macrophyte growing normally to a height of 0.5 m to 1 m. This weed causes, substantial economic and ecological harm, warranting its removal, lest the impacts of the weed should become permanent. From an ecological perspective, the most effective, management approach is to make use of the weed for various purposes. The present study o...

Abstract We prove Turán-type theorems for two related Ramsey problems raised by Bollobás and Fox Sudakov. First, t ≥ 3, we show that any two-colouring of the complete graph on n vertices is δ -far from being monochromatic contains an unavoidable t-colouring when ≫ −1/ , where -colouring a clique order 2 in which one colour forms either or disjoint cliques . Next, tournament transitive t-tournam...

Let G = (V,E) be an undirected graph without loops and multiple edges [1], V (G) and E(G) be the sets of vertices and edges of G, respectively. The degree of a vertex x ∈ V (G) is denoted by dG(x), the maximum degree of a vertex of G-by ∆(G), and the chromatic index [2] of G-by χ(G). A graph is regular, if all its vertices have the same degree. If α is a proper edge colouring of the graph G [3]...

Superficially coloured lead crystal glass was obtained by ion-exchange with silver. Both red and yellow ruby colouring were achieved by the presence of small silver particles (colloids) dispersed into the ion-exchanged glass surface. Direct silver ion-exchange of the glass yielded yellow colouring (optical absorption around 410 nm). Silver ion-exchange of lead glass previously doped with a redu...

Graph colouring enjoys many practical as well theoretical uses. is still a very active subject of research. This article introduces new concept the chromatic number neutrosophic graph (NG). There discussion difference between fuzzy (FG) and NG. Finally, real-life application for detecting phishing websites has been demonstrated by NG colouring.

Restricted star colouring is a variant of introduced to design heuristic algorithms estimate sparse Hessian matrices. For $k\in\mathbb{N}$, $k$-restricted ($k$-rs colouring) graph $G$ function $f:V(G)\to{0,1,\dots,k-1}$ such that (i)$f(x)\neq f(y)$ for every edge $xy$ G, and (ii) there no bicoloured 3-vertex path ($P_3$) in with the higher colour on its middle vertex. We show $k\geq 3$, it NP-c...

In this paper, we study a Ramsey-type problem for equations of the form ax+by=p(z). We show that if certain technical assumptions hold, then any 2-colouring positive integers admits infinitely many monochromatic solutions to equation This entails 2-Ramseyness several notable cases such as ax+y=zn arbitrary a∈Z+ and n≥2, also ax+by=aDzD+…+a1z∈Z[z] gcd(a,b)=1, D≥2, a,b,aD>0 a1≠0.