(s,m)-radius of k-connected graphs

The ( s , m ) -radius ρ s , m ( G ) is an important parameter associated with a message transmission network G . In [H. Li, J.P. Li, ( s , m ) -radius in connected graphs I, Discrete Appl. Math. (resubmitted). Also see the Ph.D. thesis 5779, Université de Paris Sud, France] we obtained upper bounds on ρ s , m ( G ) for all given integers s and m . The purpose of this paper is to present improved upper bounds on ρ s , m ( G ) for m = 3 , 4 . To be specific, for any k -connected graph G , we show that ρ s , 3 ( G ) ≤ max { 2 n s , min { 3 n s , n k + 5 n k s } + n s } if k ≥ 3 and ρ s , 4 ( G ) ≤ min { 3 n s , n k − 1 + 5 n s ( k − 1 ) } + 2 n s if k ≥ 4 .