On Graphs With Prescribed Chromatic Number and Subset Index

For a nontrivial graph G , a subset labeling of G is a labeling of the vertices of G with nonempty subsets of the set [ r ] = { 1 , 2 , . . . , r } for a positive integer r such that two vertices of G have disjoint labels if and only if the vertices are adjacent. The subset index of G is the minimum positive integer r for which G has such a subset labeling from the set [ r ] . Structures of graphs with prescribed subset index are investigated. It is shown that for every two integers a and b with 2 ≤ a ≤ b , there exists a connected graph with chromatic number a and subset index b .