In this paper, we investigate a suborbital graph for the normalizer of Γ0(N) ∈ PSL(2;R), where N will be of the form 24p2 such that p > 3 is a prime number. Then we give edge and circuit conditions on graphs arising from the non-transitive action of the normalizer.
The investigation on the locating-chromatic number of a graph was initiated by Chartrand et al. (2002). This concept is in fact a special case of the partition dimension of a graph. This topic has received much attention. However, the results are still far from satisfaction. We can define the locating-chromatic number of a graph G as the smallest integer k such that there exists a k-partition of the vertex-set of G such that all vertices have distinct coordinates with respect to this partition. As we know that the metric dimension of a tree is completely solved. However, the locating-chromatic numbers for most of trees are still open. For i = 1, 2, . . . , t, let Ti be a tree with a fixed edge eoi called the terminal edge. The edge-amalgamation of all Tis denoted by Edge-Amal{Ti;eoi} is a tree formed by taking all the Tis and identifying their terminal edges. In this paper, we study the locating-chromatic number of the edge-amalgamation of arbitrary trees. We give lower and upper bounds for their locating-chromatic numbers and show that the bounds are tight.
Burger and Vuuren defined the size multipartite Ramsey number for a pair of complete, balanced, multipartite graphs mj(Kaxb,Kcxd), for natural numbers a,b,c,d and j, where a,c >= 2, in 2004. They have also determined the necessary and sufficient conditions for the existence of size multipartite Ramsey numbers mj(Kaxb,Kcxd). Syafrizal et al. generalized this definition by removing the completeness requirement. For simple graphs G and H, they defined the size multipartite Ramsey number mj(G,H) as the smallest natural number t such that any red-blue coloring on the edges of Kjxt contains a red G or a blue H as a subgraph. In this paper, we determine the necessary and sufficient conditions for the existence of multipartite Ramsey numbers mj(G,H), where both G and H are non complete graphs. Furthermore, we determine the exact values of the size multipartite Ramsey numbers mj(K1,m, K1,n) for all integers m,n >= 1 and j = 2,3, where K1,m is a star of order m+1. In addition, we also determine the lower bound of m3(kK1,m, C3), where kK1,m is a disjoint union of k copies of a star K1,m and C3 is a cycle of order 3.
A G-decomposition of the complete graph Kn is a family of pairwise edge disjoint subgraphs of Kn, all isomorphic to G, such that every edge of Kn belongs to exactly one copy of G. Using standard decomposition techniques based on ρ-labelings, introduced by Rosa in 1967, and their modifications we show that each of the ten non-isomorphic connected unicyclic graphs with eight edges containing the pentagon decomposes the complete graph Kn whenever the necessary conditions are satisfied.
An oriented k − coloring of an oriented graph G⃗ is a partition of V(G⃗) into k color classes such that no two adjacent vertices belong to the same color class, and all the arcs linking the two color classes have the same direction. The oriented chromatic number of an oriented graph G⃗ is the minimum order of an oriented graph H⃗ to which G⃗ admits a homomorphism to H⃗. The oriented chromatic number of an undirected graph G is the maximum oriented chromatic number of all possible orientations of the graph G. In this paper, we show that every edge amalgamation of cycle graphs, which also known as a book graph, has oriented chromatic number less than or equal to six.
For some ordered subset W = {w1, w2, ⋯, wt} of vertices in connected graph G, and for some vertex v in G, the metric representation of v with respect to W is defined as the t-vector r(v∣W) = {d(v, w1), d(v, w2), ⋯, d(v, wt)}. The set W is the resolving set of G if for every two vertices u, v in G, r(u∣W) ≠ r(v∣W). The metric dimension of G, denoted by dim(G), is defined as the minimum cardinality of W. Let G be a connected graph on n vertices. The thorn graph of G, denoted by Th(G, l1, l2, ⋯, ln), is constructed from G by adding li leaves to vertex vi of G, for li ≥ 1 and 1 ≤ i ≤ n. The subdivided-thorn graph, denoted by TD(G, l1(y1), l2(y2), ⋯, ln(yn)), is constructed by subdividing every li leaves of the thorn graph of G into a path on yi vertices. In this paper the metric dimension of thorn of complete graph, dim(Th(Kn, l1, l2, ⋯, ln)), li ≥ 1 are determined, partially answering the problem proposed by Iswadi et al . This paper also gives some conjectures for the lower bound of dim(Th(G, l1, l2, ⋯, ln
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