document: Arranging solid balls to represent a graph By solid balls , we mean a set of balls in R/sup 3 / no two of which can penetrate each other . Every finite graph G can be represented by arranging solid balls in the following way : Put red balls in R/sup 3 / , one for each vertex of G , and connect two red balls by a chain when they correspond to a pair of adjacent vertices of G , where a chain means a finite sequence of blue solid balls in which each consecutive balls are tangent . -LRB- We may omit the chain if the two red balls are already tangent . -RRB- The ball number b -LRB- G -RRB- of G is the minimum number of balls -LRB- red and blue -RRB- necessary to represent G . If we put the balls and chains on a table so that all balls sit on the table , then the minimum number of balls for G is denoted by bT -LRB- G -RRB- . Among other things , we prove that b -LRB- K/sub 6 / -RRB- = 8 , b -LRB- K/sub 7 / -RRB- = 13 and b/sub T / -LRB- K/sub 5 / -RRB- = 8 , b/sub T / -LRB- K/sub 6 / -RRB- = 14 . We also prove that c/sub 1/n/sup 3 / < b -LRB- K/sub n / -RRB- < c/sub 2/n/sup 3 / log n , c/sub 3/n/sup 4 / / log n < b/sub T / -LRB- K/sub n / -RRB- < c/sub 4/n/sup 4 /
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extractive_keyphrases: solid balls finite graph adjacent vertices finite sequence
abstractive_keyphrases: graph representation