On Legendre Cordial Labeling of Some Graphs

For a simple connected graph G = (V, E), a bijective function f : V(G) → {1, 2, ..., |V|} is said to be a Legendre cordial labeling modulo p, where p is an odd prime, if the induced function fp ∗ : E(G) → {0, 1} defined by fp ∗ (uv) = 0 if (ωuv/p) = −1 or ωuv = 0, and fp ∗(uv) = 1 if (ωuv/p) = 1, where f (u) + f (v) ≡ ωuv (mod p), 0 ≤ ωuv < p, satisfies the condition efp * (0) − efp ∗ (1) ≤ 1, where efp ∗ (i) is the number of edges with label i (i = 0, 1). In this paper, we determined the values of n, m, and k so that the following special graphs admit a Legendre cordial labeling modulo p for an odd prime p: path graph Pn, cycle graph Cn, star graph Sn, tadpole graph Tn,m, kayak paddle graph KPn,m,k, bistar graph Bn,m, and fan graph Fn.