A. N. M. Salman

On the total vertex irregularity strength of trees

A vertex irregular total k-labelling @l:V(G)@?E(G)@?{1,2,...,k} of a graph G is a labelling of vertices and edges of G done in such a way that for any different vertices x and y, their weights wt(x) and wt(y) are distinct. The weight wt(x) of a vertex x is the sum of the label of x and the labels of all edges incident with x. The minimum k for which a graph G has a vertex irregular total k-labelling is called the total vertex irregularity strength of G, denoted by tvs(G). In this paper, we determine the total vertex irregularity strength of trees.

The metric dimension of the lexicographic product of graphs

Abstract A set of vertices W resolves a graph G if every vertex is uniquely determined by its coordinate of distances to the vertices in W . The minimum cardinality of a resolving set of G is called the metric dimension of G . In this paper, we consider a graph which is obtained by the lexicographic product between two graphs. The lexicographic product of graphs G and H , which is denoted by G ∘ H , is the graph with vertex set V ( G ) × V ( H ) = { ( a , v ) | a ∈ V ( G ) , v ∈ V ( H ) } , where ( a , v ) is adjacent to ( b , w ) whenever a b ∈ E ( G ) , or a = b and v w ∈ E ( H ) . We give the general bounds of the metric dimension of a lexicographic product of any connected graph G and an arbitrary graph H . We also show that the bounds are sharp.

Path-fan Ramsey numbers

For two given graphs F and H, the Ramsey number R(F, H) is the smallest positive integer p such that for every graph G on p vertices the following holds: either G contains F as a subgraph or the complement of G contains H as a subgraph. In this paper, we study the Ramsey numbers R(Pn, Fm), where Pn is a path on n vertices and Fm is the graph obtained from m disjoint triangles by identifying precisely one vertex of every triangle (Fm is the join of K1 and mK2). We determine the exact values of R(Pn, Fm) for the following values of n and m 1 ≤n≤5 and m≥2;n≥6 and 2≤m≤(n+1)/2;6≤n≤7 and m≥n-1;n≥8 and n-1≤m≤n or ((q.n - 2q + 1)/2 ≤ m ≤ (q.n-q+2)/2 with 3 ≤q ≤ n-5) orm ≥ (n-3)2/2;oddn ≥ 9 and ((q.n-3q+1)/2 ≤ m ≤(q.n-2q)/2 with 3 ≤ q ≤ (n - 3) / 2) or ((q.n - q - n + 4) / 2 ≤ m ≤ (q.n - 2q) / 2 with (n - 1) / 2 ≤ q ≤ n - 5). Moreover, we give nontrivial lower bounds and upper bounds for R(Pn, Fm) for the other values of m and n.

Backbone colorings along stars and matchings in split graphs: their span is close to the chromatic number ∗

We continue the study on backbone colorings, a variation on classical vertex colorings that was introduced at WG2003. Given a graph

Path-kipas Ramsey numbers

G = (V,E)

Supermagic coverings of the disjoint union of graphs and amalgamations

and a spanning subgraph

Improved Upper Bounds for λ-Backbone Colorings Along Matchings and Stars

H

Total Irregularity Strength of Three Families of Graphs

of

Rainbow connection number of rocket graphs

G

A characterization of the corona product of a cycle with some graphs based on its f-chromatic index

(the backbone of