On the Q(a)P(b)-Super Edge-Graceful (p,p+1)-Graphs

摘要

Let a and b be two positive integers. For the graph G with vertex set V(G) and edge set E(G) with p=|V(G)| and q=|E(G)|, we define two sets Q(a) and P(b) as follows:rnQ(a) ={ ± a, ± (a+1),..., ±(a +(q-2)/2)} if q is even, Q(a) = {0} ∪{ ± a, ± (a+1),..., ±(a +(q-3)/2)} if q is odd, P(b)={ ±b, ±(b+1),..., ±(b+(p-2)/2)} if pis even, P(b) = {0} ∪{ ±b, ±(b+1),..., ±(b+(p-3)/2)} if pis odd. For the graph G with p=|V(G)| and q=|E(G)|, G is said to bernQ(a)P(b)-super edge-graceful (in short Q(a)P(b)-SEG), if there exists a function pair (f, f~+) which assigns integer labels to the vertices and edges; that is, f~+: V (G) → P(b), and f: E (G) → Q(a) such that f~+ is onto P(b) and f is onto Q(a), and f~+(u) = Σ{f(u,v): (u, v) ∈ E (G) }. We investigate Q(a)P(b) super-edge-graceful labelings for three classes of (p,p+1)- graphs.