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Let G=(V,E) be a (p,q)-graph without isolated vertices. The gracefulness grac(G) of G is the smallest positive integer k for which there exists an injective function f:V→{0,1,2,…,k} such that the edge induced function gf:E→{1,2,…,k} defined by gf(uv)=|f(u)−f(v)| is also injective. Let c(f)=max{i:1,2,…,iare edge labels} and let m(G)=maxf{c(f)} where the maximum is taken over all injective functions...
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