The feasibility problem for line graphs

Abstract

We consider the following feasibility problem: given an integer n ≥ 1 and an integer m such that 0 ≤ m ≤ n 2 , does there exist a line graph L = L ( G ) with exactly n vertices and m edges? We say that a pair ( n , m ) is non-feasible if there exists no line graph L ( G ) on n vertices and m edges, otherwise we say ( n , m ) is a feasible pair. Our main result shows that for fixed n ≥ 5 , the values of m for which ( n , m ) is a non-feasible pair form disjoint blocks of consecutive integers which we determine completely. On the other hand we prove, among other things, that for the more general family of claw-free graphs (with no induced K 1 , 3 -free subgraph), all ( n , m ) -pairs in the range 0 ≤ m ≤ n 2 are feasible pairs.