Edge number of 3-connected diameter 3 graphs

Abstract

Let the decay number, /spl zeta/(G) be the minimum number of components of a cotree of a connected graph G. Let /spl Omega/ be the collection of all 3-connected diameter 3 graphs. In this paper, we prove that if k is the minimum number such that q /spl ges/ 2p - k for each (p,q)-graph G /spl epsi/ /spl Omega/, and 1 is the minimum number such that /spl zeta/(H) /spl les/ l - 1 for each graph H /spl epsi/ /spl Omega/, then k=l. Furthermore, we prove that k /spl les/ 11 and we find a 3-connected, diameter 3 graph with q = 2p - 8. So we have that 8 /spl les/ k /spl les/ 11 and we conjecture that k = 8.