Regular Honest Graphs, Isoperimetric Numbers, and Bisection of Weighted Graphs

The edge-integrity of a graph G is I 0(G) := minfjSj + m(G S) : S Eg; where m(H) denotes the maximum order of a component of H: A graph G is called honest if its edge-integrity is the maximum possible, that is, equals the order of the graph. The only honest 2-regular graphs are the 3-, 4-, and 5-cycles. Lipman [13] proved that there are exactly twenty honest cubic graphs. In this paper we exploit a technique of Bollob as [8, 9] to prove that for every k 6, almost all k-regular graphs are honest. On the other hand, we show that there are only nitely many 4-regular honest graphs. To prove this, we use a weighted version of the upper bound on the isoperimetric number due to Alon [1]. We believe that this version is of interest by itself. Research supported in part by a USA Israeli BSF grant and by the Fund for Basic Research administered by the Israel Academy of Sciences. This work was partially supported by an Indiana-University Purdue-University Fort Wayne Scholar-inResidence Grant, and by the grant 97-01-01075 of the Russian Foundation for Fundamental Research.