The feasibility problem for line graphs

TL;DR

This paper investigates which pairs of vertices and edges can be realized by line graphs, providing a complete characterization of non-feasible pairs for fixed vertex counts and showing all pairs are feasible for claw-free graphs.

Contribution

It characterizes the non-feasible pairs for line graphs with fixed vertices and proves all pairs are feasible for claw-free graphs.

Findings

Non-feasible pairs form disjoint blocks of consecutive integers for fixed n.

Complete characterization of non-feasible pairs for line graphs with n ≥ 5.

All pairs are feasible in the class of claw-free graphs.

Abstract

We consider the following feasibility problem: given an integer $n \geq 1$ and an integer $m$ such that $0 \leq m \leq \binom{n}{2}$, 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 $n\geq 5$, the values of $m$ for which $(n, m)$ is a non-feasible pair, form disjoint blocks of consecutive integers which we completely determine. 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 $0 \leq m \leq \binom{n}{2}$ are feasible pairs.