Odd harmonious labeling of grid graphs

摘要

A graph G(p, q) is said to be odd harmonious if there exists an injection f : V (G) → {0, 1, 2, · · · , 2q ? 1} such that the induced function f* : E(G) → {1, 3, · · · , 2q ? 1} defined by f? (uv) = f (u) + f (v) is a bijection. In this paper we prove that path union of t copies of Pm×Pn, path union of t different copies of Pm?×Pn? where 1 ≤ i ≤ t, vertex union of t copies of Pm×Pn, vertex union of t different copies of Pm?×Pn? where 1 ≤ i ≤ t, one point union of path of Ptn (t.n.Pm×Pm), t super subdivision of grid graph Pm×Pn are odd harmonious graphs.