Graph 2-rankings

Asymptotically Optimal Vertex Ranking of Planar Graphs

“…Shalu and Antony [30] show that determining the minimum number of colours required by a 2-ranking of a given graph is NP-hard, even when restricted to planar bipartite graphs. Almeter et al [1] determine the exact value of χ 2 (Q d ) = d + 1 where Q d is the d-cube. They also shows that, for graphs G of maximum degree 3, χ 2 (G) ≤ 7 and show the existence of a graph with maximum degree k such that χ 2 (G) ∈ Ω(k 2 / log k).…”

“…Globally, the proof is by induction on the value of t, though it is easy to miss this, since it is spread over several lemmas. The case t = 1 is easy: By Lemma 12(i), any graph of simple treewidth 1 is a contained in a path and therefore has an ℓ-ranking using ℓ + 1 ∈ O(log n/ log (1) n) = O(1) colours. In the proof of Lemma 29, below, we will apply Theorem 2a to graphs of simple treewidth t − 1.…”

The complexity of restricted star colouring

“…for definitions). It was also studied independently in the guise of unique superior colouring [4,5], a generalization of ordered colouring (see Section 6 for the definition of ordered colouring). Therefore, restricted star colouring (abbreviated rs colouring) is intermediate in strength between star colouring and ordered colouring.…”

“…Restricted star colouring is studied in the literature mainly for minor-excluded graph families [4], trees [4], q-degenerate graphs [4], and degree-bounded graph families [5]. The rs chromatic number of a graph G, denoted by χrs(G), is the least integer k such that G is k-rs colourable.…”

“…For some classes such as planar graphs and trees, only asymptotic bound for the rs chromatic number is known [4]. On the other hand, there are classes for which we can compute the rs chromatic number in polynomial time using a simple formula; for instance, χrs(Q d ) = d + 1 for the hypercube Q d [5]. It is also known that χrs(G) ≤ 4α(G) for every graph G of girth at least five [10].…”