On the edge irregularity strength for some classes of plane graphs

: Graph labeling is an assignment of (usually) positive integers to elements of a graph (vertices and / or edges) satisfying certain condition(s). In the last two decades, graph labeling research received much attention from researchers. This articles is about edge irregularity strength for some classes of plane graphs. Edge irregularity strength denoted by es ( G ), was introduced by Ahmad et al. in 2014 as a modiﬁcation of the well known irregularity strength by Chartrand in 1988. In this paper, the exact value of the edge irregularity strength for some clases of plane graphs is determined.


Introduction
Let G be a connected, simple and undirected graph with vertex set V(G) and edge set E(G). Graph labeling is a mapping of elements of the graph, i.e. vertex and/or edges to a set of numbers (usually positive integers), called labels. If the domain is the vertex-set or the edge-set, the labeling is called vertex labeling or edge labeling respectively. Similarly if the domain is V(G) ∪ E(G), then the labeling is called total labeling. In 1988, Chartrand et al. [19] defined irregular labeling for a graph G as an assignment of labels from the set of natural numbers to the edges of G such that the sums of the labels assigned to the edges of each vertex are different. The minimum value of the largest label of an edge over all existing irregular labelings is known as the irregularity strength of G, and it is denoted by s(G). The work of Chartrand et al. [19] opened a new horizon for graph theorists with a lot of research in this domain as confessed by the numerous articles investigating s(G) for various families of graphs (see [7,10,18,20,21,26,27]).
In 2007, Baca et al. in [15] investigated two modifications of the irregularity strength of graphs, namely total edge irregularity strength denoted by tes(G), and total vertex irregularity strength denoted by tvs(G). Results on the total vertex irregularity strength and the total edge irregularity strength can be found in [1,2,4,8,11,16,23,25,[27][28][29][30]. Motivated by the work of Chartrand et al. [19], Ahmad et al. in [3] introduced edge irregular klabelings of graphs. A vertex k-labeling of graph G φ : V(G) → {1, 2, · · · , k} can be defined as an edge irregular k-labeling for G if for every two different edges e and f it is w φ (e) w φ ( f ), where the weight w φ (e) of an edge e = xy ∈ E(G) is defined as w φ (xy) = φ(x) + φ(y). The minimum k for which the graph G has an edge irregular k−labeling is called the edge irregularity strength of G, denoted by es(G). In the same work [3], the authors proved a general lower bound of es(G) and then determined exact values for several families of graphs such as paths, stars, double stars and Cartesian product of two paths. Over the last years, es(G) has been investigated for different families of graphs including trees with the help of algorithmic solutions and amalgamated families of graph through corona product and Toeplitz graphs [5,6,9,[12][13][14][31][32][33][34][35].
A lot of work has focused on graph labeling and that is evident from the recent survey by Gallian [22]. Still there is great potential of expansion in this area. That is why in this paper, we determine the exact value of edge irregularity strength for some classes of plane graphs. A planar graph is a graph that can be drawn on the plane in such a way that its edges do no intersect and only meet at their endpoints. A plane graph is a particular drawing of a planar graph on the Euclidean plane.

Main results
The following theorem establishes a general lower bound for the edge irregularity strength of a graph G (see [3]).
We first discuss edge irregularity strength of plane graph C n that is defined in [17] as follows: Let P 1 , P 2 and P 3 be paths on vertices a 1 , a 2 , . . . , a n ; b 1 , b 2 , . . . , b 2n and c 1 , c 2 , . . . , c n , respectively. Form the graph C n from the disjoint union P 1 ∪ P 2 ∪ P 3 by adding the edges Graph C n is shown in Figure 1.
In the following theorem, we determine the exact value of the edge irregularity strength of C n .
Proof. Let C n be the graph with vertex set V( The order and size of graph C n is 4n and 6n − 3, respectively. From Theorem 1, we have es(C n ) ≥ max 6n−2 To prove the equality, it suffices to prove the existence of an optimal edge irregular (3n − 1)-labeling.
We construct the labeling ψ 1 : V(C n ) → {1, 2, 3, . . . , 3n − 1} in the following way: The edge weights are as follows: We can see that all vertex labels are at most 3n − 1. The edge weights under the labeling ψ 1 are distinct for all pairs of distinct edges and the labeling ψ 1 provides the upper bound on es(C n ), i.e es(C n ) ≤ 3n − 1. Combining with the lower bound, we conclude that es(C n ) = 3n − 1. This completes the proof.
Let A n denotes the plane graph consisting of faces of length s where s = 3, 4, 5 and one external infinite face. Note that graph A n is similar to C n if one adds edges Moreover |V(A n )| = 4n and |E(A n )| = 7n − 4, see [11,24]. The graph A n is shown in Figure 2.
In the following theorem, we determine the exact value of the edge irregularity strength of A n .  To prove the equality, it suffices to prove the existence of an optimal edge irregular 7n−3 2 -labeling. Let The edge weights are as follows: We can see that all vertex labels are at most 7n A quadrilateral snake Q n is a plane graph obtained from a path b 1 , b 2 , . . . , b n by adding new vertices a 1 , a 2 , a 3 . . . , a 2(n−1) and new edges a 2i−1 a 2i , respectively and joining a 2i−1 to b − i and a 2i to b i+1 . The double quadrilateral snake D(Q n ) is obtained from Q n by adding new vertices c 1 , c 2 , . . . , c 2(n−1) and new edges c 2i−1 c 2i , c 2i−1 b i and c 2i b i+1 for 1 ≤ i ≤ n − 1. It is clear that |V(D(Q n ))| = 5n − 4 and |E(D(Q n ))| = 7(n − 1). The graph D(Q n ) is shown in Figure 3.  Proof. Let D(Q n ) be the plane graph with vertex set V(D(Q n )) = {a i , c i : The maximum degree of D(Q n ) is ∆(D(Q n )) = 6. From Theorem 1, we have that es(D(Q n )) ≥ max 7n−6 2 , 6 = 7n−6 2 .
To prove the equality, it suffices to prove the existence of an optimal edge irregular 7n−6 2 -labeling.
Let ψ 3 : V(D(Q n )) → {1, 2, . . . , 7n−6 2 be the vertex labeling such that The edge weight are as follows: We can see that all vertex labels are at most 7n−6 2 . The edge weights under the labeling ψ 3 are distinct for all pairs of distinct edges and the labeling ψ 3 provides the upper bound on es(D(Q n )), i.e. es(D(Q n )) ≤ 7n−6 2 . Combining with the lower bound, we conclude that es(D(Q n )) = 7n−6 2 . This completes the proof.

Remarks and Conclusion
The problem studied in this paper is about edge irregularity strength of three classes of plane graphs. According to result for lower bound of es(G) in Theorem 1 and all three upper bounds in Theorems 2, 3 and 4, we obtain the exact value for edge irregularity strength of these graphs.
The graphs considered in this paper are quite restricted. From our understanding, the es(G) is indeed hard to compute for general graph G, or even for planar graphs. As families of planar graphs, we expect to study outerplanar graphs, planar graphs with bounded maximum degree, or with faces of small size, planar 2-trees, planar 3-trees, etc. They are common graph families in the literature of graph labeling and provide insight for other families of planar graphs that include them.
Presented graphs have bounded tree-width and path-width. Although this might be a mere observation, one can find a path decomposition such that the induced subgraphs in each partitions are identical (except possibly for the last one) and have E n edges. For example, for D(Q n ) the path decomposition is defined by associating the 2-connected components to the nodes of the path. Two consecutive subgraphs have one vertex in common (cutvertex) and each subgraph has 7 edges (recall that E(D(Q n )) = 7n − 7. One can compute the labeling for the first subgraph and propagate by linearly increasing the labels. If the edge irregularity is satisfied for the first two subgraphs, then it should hold for the entire graph. Such an approach (if correct) could potentially broaden the targeted graphs.