Bound on the irregularity strength of regular graphs

Indonesian Journal of Combinatorics, 2021

An edge irregular total k-labeling f : V ∪ E → {1, 2, ..., k} of a graph G = (V, E) is a labeling of vertices and edges of G in such a way that for any two different edges uv and u v , their weights f (u) + f (uv) + f (v) and f (u) + f (u v) + f (v) are distinct.The total edge irregularity strength tes(G) is defined as the minimum k for which the graph G has an edge irregular total k-labeling. In this paper, we determine the total edge irregularity strength of new classes of graphs C m @C n , P * m,n and C * m,n and hence we extend the validity of the conjecture tes(G) = max |E(G)|+2 3 , ∆(G)+1 2 for some more graphs.

Journal of Graph Theory, 2002

An assignment of positive integer weights to the edges of a simple graph G is called irregular if the weighted degrees of the vertices are all different. The irregularity strength, s(G), is the maximal weight, minimized over all irregular assignments. In this paper we show, that s(G) ≤ c 1 n/δ, for graphs with maximum degree ∆ ≤ n 1/2 and minimum degree δ, and s(G) ≤ c 2 (log n)n/δ, for graphs with ∆ > n 1/2 , where c 1 and c 2 are explicit constants. To prove the result, we are using a combination of deterministic and probabilistic techniques.

A totally irregular total k-labeling

Discrete Applied Mathematics, 1987

Indonesian Journal of Combinatorics, 2020

Under a totally irregular total k-labeling of a graph G = (V, E), we found that for some certain graphs, the edge-weight set W (E) and the vertex-weight set W (V) of G which are induced by k = ts(G), W (E) ∩ W (V) is a nonempty set. For which k, a graph G has a totally irregular total labeling if W (E) ∩ W (V) = ∅? We introduce the total disjoint irregularity strength, denoted by ds(G), as the minimum value k where this condition satisfied. We provide the lower bound of ds(G) and determine the total disjoint irregularity strength of cycles, paths, stars, and complete graphs.

Annals of the New York Academy of Sciences, 1989

Ars Combinatoria - ARSCOM, 2004

The domatic number of a graph G is the maximum number of dominating sets into which the vertex set of G can be partitioned. We show that the domatic number of a random r-regular graph is almost surely at most r, and that for 3-regular random graphs, the domatic number is almost surely equal to 3. We also give a lower bound on the domatic number of a graph in terms of order, minimum degree and maximum degree. As a corol- lary, we obtain the result that the domatic number of an r-regular graph is at least (r + 1)/(3ln(r + 1)).

Filomat, 2014

Albertson [3] has defined the P irregularity of a simple undirected graph G = (V,E) as irr(G) =?uv?E |dG(u)- dG(v)|, where dG(u) denotes the degree of a vertex u ? V. Recently, this graph invariant gained interest in the chemical graph theory, where it occured in some bounds on the first and the second Zagreb index, and was named the third Zagreb index [12]. For general graphs with n vertices, Albertson has obtained an asymptotically tight upper bound on the irregularity of 4n3/27: Here, by exploiting a different approach than in [3], we show that for general graphs with n vertices the upper bound ?n/3? ?2n/3? (?2n/3? -1) is sharp. We also present lower bounds on the maximal irregularity of graphs with fixed minimal and/or maximal vertex degrees, and consider an approximate computation of the irregularity of a graph.

Graphs and Combinatorics, 1992

Let r be a positive integer. Consider r-regular graphs in which no induced subgraph on four vertices is an independent pair of edges. The number v of vertices in such a graph does not exceed 5r/2; this proves a conjecture of Bermond. More generally, it is conjectured that if v > 2r, then the ratio v/r must be a rational number of the form 2 + 1/(2k). This is proved for v/r > 21 _~. The extremal graphs and many other classes of these graphs are described and characterized.

Discussiones Mathematicae Graph Theory, 2019

For a graph G an edge-covering of G is a family of subgraphs H 1 , H 2 ,. .. , H t such that each edge of E(G) belongs to at least one of the subgraphs H i , i = 1, 2,. .. , t. In this case we say that G admits an (H 1 , H 2 ,. .. , H t)-(edge) covering. An H-covering of graph G is an (H 1 , H 2 ,. .. , H t)-(edge) covering in which every subgraph H i is isomorphic to a given graph H. Let G be a graph admitting H-covering. An edge k-labeling α : E(G) → {1, 2,. .. , k} is called an H-irregular edge k-labeling of the graph G if for every two different subgraphs H ′ and H ′′ isomorphic to H their weights 1 Corresponding author. 2 Naeem, Siddiqui, Bača, Semaničová-Feňovčíková and Ashraf wt α (H ′) and wt α (H ′′) are distinct. The weight of a subgraph H under an edge k-labeling α is the sum of labels of edges belonging to H. The edge H-irregularity strength of a graph G, denoted by ehs(G, H), is the smallest integer k such that G has an H-irregular edge k-labeling. In this paper we determine the exact values of ehs(G, H) for prisms, antiprisms, triangular ladders, diagonal ladders, wheels and gear graphs. Moreover the subgraph H is isomorphic to only C 4 , C 3 and K 4 .