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

The (s,m)(s,m)-radius ρs,m(G)ρs,m(G) is an important parameter associated with a message transmission network GG. In [H. Li, J.P. Li, (s,m)(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 ρs,m(G)ρs,m(G) for all given integers ss and mm. The purpose of this paper is to present improved upper bounds on ρs,m(G)ρs,m(G) for m=3,4m=3,4. To be specific, for any kk-connected graph GG, we show that ρs,3(G)≤max{2ns,min{3ns,nk+5nks}+ns} if k≥3k≥3 and ρs,4(G)≤min{3ns,nk−1+5ns(k−1)}+2ns if k≥4k≥4.