Colorings of complements of line graphs

Abstract

Our purpose is to show that complements of line graphs (of graphs) enjoy nice coloring properties. We show that for all graphs in this class the local and usual chromatic numbers are equal. We also prove a sufficient condition for the chromatic number to be equal to a natural upper bound. A consequence of this latter condition is a complete characterization of all induced subgraphs of the Kneser graph KG ( n , 2 ) that have a chromatic number equal to its chromatic number, namely n − 2. In addition to the upper bound, a lower bound is provided by Dol'nikov's theorem, a classical result of the topological method in graph theory. We prove the NP‐hardness of deciding the equality between the chromatic number and any of these bounds. The topological method is especially suitable for the study of coloring properties of complements of line graphs of hypergraphs. Nevertheless, all proofs in this article are elementary and we also provide a short discussion on the ability for the topological methods to cover some of our results.

Local orthogonality dimension

“…, the theorem implies that for complements of lines graphs, the chromatic number is equal to its local variant, and thus strengthens the result of [8] given in Theorem 1.2.…”

“…This shows that there are topologically $$t$$‐chromatic graphs $$G$$ with $$\chi (G)=t$$ for which the lower bound given in Theorem 1.5 has a multiplicative gap of roughly 2 from the truth. Additionally, since every graph $$G$$ satisfies $${\bar{\xi }}_{l}(G,{\mathbb{R}})\le {\chi }_{l}(G)\le \chi (G)$$, the theorem implies that for complements of lines graphs, the chromatic number is equal to its local variant, and thus strengthens the result of [8] given in Theorem 1.2.…”

“…≥ , hence the upper bound of t is tight in this case. This result was recently generalized by Daneshpajouh, Meunier, and Mizrahi [8], as follows.…”

“…On the other hand, for the Schrijver graphs $$S(n,2)$$, it was shown in [29, Section 4.2] that the local chromatic number is equal to the chromatic number, that is, $${\chi }_{l}(S(n,2))=\chi (S(n,2))=n-2$$ for all $$n\ge 4$$, hence the upper bound of $$t$$ is tight in this case. This result was recently generalized by Daneshpajouh, Meunier, and Mizrahi [8], as follows.…”

“…, as required. To prove (8), recall that the vectors assigned to the 2-subsets of C in  include two vectors that are not proportional. In particular, it follows that  includes either two or three 2-subsets of C, and we consider the following two cases accordingly.…”