Let F, G, and H be simple graphs. The notation F → (G, H) means that if all the edges of F are arbitrarily colored by red or blue, then there always exists either a red subgraph G or a blue subgraph H. The size Ramsey number of graph G and H, denoted byr(G, H) is the smallest integer k such that there is a graph F with k edges satisfying F → (G, H). In this research, we will study a modified size Ramsey number, namely the connected size Ramsey number. In this case, we only consider connected graphs F satisfying the above properties. This connected size Ramsey number of G and H is denoted byr c (G, H). We will derive an upper bound ofr c (nK 2 , H), n ≥ 2 where H is 2P m or 2K 1,t , and find the exact values ofr c (nK 2 , H), for some fixed n.
Let G be a graph, modular k-coloring, k > 2 on graph G without isolated vertex is a vertex coloring on graph G with elements in the set of integers modulo k, Zk satisfying the properties for every two neighboring vertex in G, the number of colors (v) from their different neigbors in Zk. The modular chromatic number mc(G) in G is the minimum k integer where there is modular k-coloring on graph G. In this article describes modular chromatic numbers on Star Graph (Sn), Caterpillar Graph , Fan Graph (Fn), Helm Graph (Hn) and Triangular Book Graph (Btn).
Keywords: Modular coloring, Modular chromatic number
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