Which graphs are pluperfect?

A graph G is pluperfect if one can add new multiple edges to obtain a multi graph M satisfying: (1) G spans M, (2) the degrees of the nodes of M are consecutive integers, (3) G and M have the same minimum degree. Our purpose is to display several families of pluperfect graphs in order to L Introduction My good friends Mehdi Behzad and Gary Chartrand wrote a note [1] titled "No graph is perfect". They did this in jest in order to parody the last line, "Nobody's perfect" of the celebrated film "Some Like It Hot". They defined a perfect graph as one whose n nodes have distinct degrees and noted that there are none! Such a multigraph is now called irregular. Theorem A. No nontrivial graph is irregular. Proof Assume there exists such a graph G with n;:::: 2 nodes. Then its degree sequence is (n 1, n 2, ... , 3, 2, 1, 0). The node of degree 0 is isolated but the node of degree n 1 is adjacent to all the other nodes, a contradiction. 0 Australasian journal of Combinatorics Z (1993), pp. 81-86 The underlying graph G(M) of a multigraph M is the spanning subgraph of M having a single edge wherever M has either a single edge or multiple edges. In general we follow the notation and terminology of [2] and [4]. Recently Chartrand and et al. proved in [3] that for each connected graph G with n ~ 3 nodes, there exists an irregular multigraph M having G as its underlying graph. 2. Pluperfect graphs We now call a connected graph G with n ~ 3 nodes and minimum degree 8 plupeTject if there exists a multigraph M with underlying graph G such that the degree sequence (J of M is consecutive and as small as possible: (J = (n + 8 1, n + 8 2, ... , 8 + 2, 8 + 1, 8) (1) Our observations are elementary but novel. Ou~ purpose is to present several pluperfect graphs in order to stimulate research on the question in the title of this note. In [6], we defined the irregularity cost ic(G) as the minimum number of new multiple edges in an irregular multigraph M with underlying graph G. Let q(G) and q(M) be the number of edges of graph G and of multigraph M, respectively. Similarly let 8(G) and 8(M) be their minimum degrees. Theorem 1. For a pluperfect graph G with spanning supermultigraph M satisfying (1), ic(G) = q(M) q(G). Proof As 8(M) = 8(G), a multigraph satisfying (1) must necessarily have the smallest possible number of new multiple edges among all irregular multigraphs M with G = G(M). 0 Question 1. If G is pluperfect does there exist a unique multigraph M satisfying (1)?