Random Minimum Length Spanning Trees in Regular Graphs

r-regular n-vertex graph G with random independent edge lengths, each uniformly distributed on (0, 1). Let mst(G) be the expected length of a minimum spanning tree. We show that mst(G) can be estimated quite accurately under two distinct circumstances. Firstly, if r is large and G has a modest edge expansion property then , where . Secondly, if G has large girth then there exists an explicitly defined constant such that . We find in particular that .

[1]  J. Steele.  Probability theory and combinatorial optimization , 1987 .

[2]  Svante Janson,et al.  The Minimal Spanning Tree in a Complete Graph and a Functional Limit Theorem for Trees in a Random Graph , 1995, Random Struct. Algorithms.

[3]  L. Lovász.  Combinatorial problems and exercises , 1979 .

[4]  Mathew D. Penrose.  Random minimal spanning tree and percolation on the N-cube , 1998, Random Struct. Algorithms.

[5]  Donald E. Knuth,et al.  The Art of Computer Programming, Volume I: Fundamental Algorithms, 2nd Edition , 1997 .

[6]  Alan M. Frieze,et al.  On the value of a random minimum spanning tree problem , 1985, Discret. Appl. Math..

[7]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[8]  Kathryn Fraughnaugh,et al.  Introduction to graph theory , 1973, Mathematical Gazette.

[9]  Brendan D. McKay,et al.  Asymptotic enumeration by degree sequence of graphs with degreeso(n1/2) , 1991, Comb..

[10]  Béla Bollobás,et al.  Random Graphs , 1985 .

[11]  C. McDiarmid,et al.  On random minimum length spanning trees , 1989 .

[12]  D. Bertsimas,et al.  The Minimum Spanning Tree Constant in Geometrical Probability and Under the Independent Model: A Unified Approach , 1992 .

[13]  M. Penrose.  Random minimal spanning tree and percolation on the N -cube , 1998 .

[14]  Donald E. Knuth,et al.  The art of computer programming: V.1.: Fundamental algorithms , 1997 .

[15]  M. Talagrand.  Concentration of measure and isoperimetric inequalities in product spaces , 1994, math/9406212.

[16]  Béla Bollobás,et al.  Exact Face-isoperimetric Inequalities , 1990, Eur. J. Comb..