New formulae for the bipartite vertex frustration and decycling number of graphs

摘要

Let ∇2(G), φ(G) and ∇(G) denote the bipartite vertex frustration, bipartite edge frustration and decycling number of a graph G, respectively. In this paper, we show that ∇2(G)=|V(G)|−max {α(G−E(B)):B is a spanning bipartite subgraph of G}, which provides a new method to estimate the bipartite vertex frustration in terms of bipartite edge frustration for graphs. Using the formula above and some known results on the bipartite edge frustration, we derive several new results on the bipartite vertex frustration, including some that are equalities. Further, we study the bipartite vertex frustration of five classes of composite graphs. Finally, we present a formula for the decycling number: ∇(G)=|V(G)|−max{α(G−E(F)):FisaspanningforstofG}. Based on this result, we determine the decycling numbers of many certain graphs. In addition, this formula gives a new way for dealing with the decycling number problem of dense graphs.