Uniquely Colourable Graphs and the Hardness of Colouring Graphs of Large Girth

Abstract

For any integer k, we prove the existence of a uniquely k-colourable graph of girth at least g on at most k12(g+1) vertices whose maximal degree is at most 5k13. From this we deduce that, unless NP=RP, no polynomial time algorithm for k-Colourability on graphs G of girth g(G)[ges ]log[mid ]G[mid ]/13logk and maximum degree Δ(G)[les ]6k13 can exist. We also study several related problems.

Colouring graphs when the number of colours is almost the maximum degree

“…But the results of the past few paragraphs suggest that for large enough ∆, perhaps it is true whenever χ(G) is sufficiently close to its upper bound of ∆(G) + 1. A construction from Embden-Weinert et al [11] shows that it is not true for some graphs with χ(G) = ∆ − ⌈ √ ∆⌉:…”

“…This is well-known to be trivial for c ≤ 2. Embden-Weinert et al [11] used their construction (see Section 1.2) to prove that for 3 ≤ c ≤ ∆−k ∆ −1, we cannot test for c-colourability of graphs with maximum degree ∆ in polytime unless P = NP . On the other hand, Theorem 5 easily implies that for every constant ∆ ≥ ∆ 0 and every c ≥ ∆ − k ∆ , there is a linear time deterministic algorithm to test whether graphs of maximum degree ∆ are c-colourable.…”

“…This is well-known to be trivial for c ≤ 2. Embden-Weinert et al [11] used their construction that we presented in Section 1.2 to prove that for 3 ≤ c ≤ ∆ − k ∆ − 1, we cannot test this in polytime unless P = NP :…”

Open Problems on Graph Coloring for Special Graph Classes

“…However, no dichotomy is known for k-Coloring restricted to graphs of maximum degree at most d, but some partial results have been obtained. Molloy and Reed [60] classified the complexity for all pairs (k, d) for sufficiently large d, whereas Emden-Weinert et al [19] showed that k-Coloring is NP-complete for graphs of maximum degree at most k + √ k − 1. By combining the latter result with Brooks' Theorem [8], we find that the smallest open case is the following problem.…”

A Combinatorial Classic — Sparse Graphs with High Chromatic Number

“…The question was refined in [86] to lattices and this paper also contains a polynomial algorithm (using Ramanujan graphs) which constructs for given k and l a graph G k,l with girth(G k,l ) = l and χ(G k,l ) = k (see also [22]). …”