Two Modulo Three Sum Graphs

ABSTRACT

Let G = (V, E) be a graph with p vertices and q edges is said to be a two modulo three sum graph if there is an injective function 𝑓 from V(G) to { 𝑎∶0≤𝑎≤3𝑞−1 and either 𝑎≡0(𝑚𝑜𝑑 3) 𝑜𝑟 𝑎≡2(𝑚𝑜𝑑 3)} where q is the number of edges of G and such that 𝑓induces a bijection𝑓∗ from E(G) to {𝑎∶2≤𝑎≤3𝑞−1 𝑎𝑛𝑑 𝑎≡2(𝑚𝑜𝑑 3)} given by𝑓∗(𝑢𝑣)=𝑓(𝑢)+𝑓(𝑣)and the function 𝑓 is called two modulo three sum labeling of G. In this paper, we introduce an analog of sum labeling known as two modulo three sum labeling and we define two modulo three sum labeling of some tree related graphs. Also we prove that split star, mirror path graph, complete bipartite graph and 𝐶4ʘ𝑛𝐾1 are two modulo three sum graphs.