Li Yusheng等人曾给出一个独立数的下界公式:α(G)≥Nfa+1(d),其中fa(x)=∫01(1-t)1/adt/(a+(x-a).t)。为了得到r(H,Kn)的上界,可以考虑建立不含H作为子图的临界图G的独立数的下界。即通过对临界图G及其邻域导出子图Gv的平均次数的分析,得出G的阶(顶点数)N与n之间的不等式关系。再利用函数fa(x)的分析性质得出当n趋于无穷大时,N+1的最小可能渐近表达式,即为r(H,Kn)的渐近上界。主要介绍这种分析方法在解决K+-K,"K+C","K"等图形和完全图Ramsey数渐近上界问题中的应用。
We briefly introduce the connection between the Shannon capacity of a communication channel and graph Ramsey number, which may receive attention from researchers on communication theory and graph theory.
LI YushengCollege of Sciences, Hohai University, Nanjing 210098, China
Let G be a graph with degree sequence ( dv). If the maximum degree of any subgraph induced by a neighborhood of G is at most m, then the independence number of G is at least $\sum\limits_v {f_{m + 1} \left( {d_v } \right)} $ , where fm+1( x) is a function greater than $\frac{{log\left( {x/\left( {m + 1} \right)} \right) - 1}}{x}for x > 0$ for x> 0. For a weighted graph G = ( V, E, w), we prove that its weighted independence number (the maximum sum of the weights of an independent set in G) is at least $\sum\limits_v {\frac{{w_v }}{{1 + d_v }}} $ where wv is the weight of v.
