In Euclidean geometry, the sum of two sides of any triangle is greater than the third side. We introduce this idea to labeling of graphs. A (p,q)-graph G=(V,E) is said to be in Euclid(0) if there exists a bijection f: V(G) –> {1,…,p} such that for each induced C3 subgraph with vertices {v1,v2,v3} with f(v1)<f(v2)<f(v3) we have

f(v1)+f(v2)>f(v3) .

For k __>__ 1, G is in Euclid(k) class of graphs if there exits smallest k such that G U Nk in Euclid(0), where Nk is the null graph with k isolated points. We exhibit infinitely many graphs in Euclid(k) for each k. The talk is target to general audiences. Several open problems will posed for future research. The report is the joint work with several high school, undergraduate students and researchers.