Decomposing graphs into interval colorable subgraphs and no-wait multi-stage schedules

A note on the interval colouring thickness of graphs

“…We write θ(n) for the maximum interval colouring thickness of a graph on n vertices, and θ ′ (m) for the maximum interval colouring thickness of a graph with m edges. Upper bounds on θ(G) in terms of the edge chromatic number of G obtained by Asratian, Casselgren, and Petrosyan [1] imply that θ(n) 2 ⌈n/5⌉, and the same authors observe that an arboricity result of Dean, Hutchinson, and Scheinerman [7] gives θ ′ (m) m/2 . Axenovich and Zheng [4] improved on the first of these bounds, showing that θ(n) is sublinear.…”

“…The bound on θ ′ (m) follows trivially (see Section 2). One corollary of this theorem is that for every integer k there exists a graph with interval colouring thickness k, answering a question of Asratian, Casselgren, and Petrosyan [1]. Indeed, given a graph G of some interval colouring thickness K, and an edge-decomposition of G into K interval colourable subgraphs, the union of any k K of these subgraphs has interval colouring thickness k.…”

“…Before proving Theorem 1, we note that it is relatively straightforward to use Theorem 5 and the bound θ ′ (m) = O ( √ m) [1,7] mentioned in the introduction to prove that 1) . Indeed, let G be an n-vertex graph and suppose that e(G) = Ω(n 2−2/13 ).…”

“…The next lemma optimises this approach for any given polynomial upper bound on θ ′ (m). 1) , where α = 10β 5+β .…”

“…In the present paper, we study a parameter introduced recently by Asratian, Casselgren, and Petrosyan [1] to quantify how far a graph is from being interval colourable. The interval colouring thickness of a graph G, denoted θ(G), is the minimum number k such that G can be edge-decomposed into k interval colourable subgraphs.…”