We prove that every connected graph with s vertices of degree not 2 has a spanning tree with at least 1/4(s- 2) + 2 leaves. Let G be a connected graph of girth g with υ vertices. Let maximal chain of successively adjacent vertices of degree 2 in the graph G does not exceed k ≥ 1. We prove that G has a spanning tree with at least αg,k(υ(G) - k - 2) + 2 leaves, where αg,k[g + 1/2]/[g + 1/2](k+3)+1 for k < g - 2; αg,k = g-2/(g-1)(k+2) for k ≥ g - 2. We present infinite series of examples showing that all these bounds are tight. Bibliography: 12 titles.
|Original language||English (US)|
|Number of pages||9|
|Journal||Journal of Mathematical Sciences (United States)|
|State||Published - Aug 2012|
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Applied Mathematics