Sum-list colouring of unions of a hypercycle and a path with at most two vertices in common

Abstract

Given a hypergraph H and a function f : V (H) −→ N, we say that H is f-choosable if there is a proper vertex colouring φ of H such that φ(v) ∈ L(v) for all v ∈ V (H), where L : V (H) −→ 2 N is any assignment of f (v) colours to a vertex v. The sum choice number Hi sc (H) of H is defined to be the minimum of v∈V (H) f (v) over all functions f such that H is f-choosable. For an arbitrary hypergraph H the inequality χ sc (H) ≤ |V (H)| + |E(H)| holds, and hypergraphs that attain this upper bound are called sc-greedy. In this paper we characterize sc-greedy hypergraphs that are unions of a hypercycle and a hyperpath having at most two vertices in common. Consequently, we characterize the hypergraphs of this type that are forbidden for the class of sc-greedy hypergraphs.

Hypergraph operations preserving sc-greediness

“…However, adding a handle preserves linearity and 2-connectivity, when the handle has at least two edges. The research on adding hypergraph handles was initiated in papers [10,11] where necessary and sufficient conditions for preserving sc-greediness were given in the case of handles added to a hypercycle. In this paper we continue this line of research by considering the operation of adding handles to arbitrary sc-greedy hypergraphs.…”

“…The following results on adding handles to hypercycles we find in [11] (see Figure 1 for an illustration).…”

“…Many of the above concepts and results related to the earlier work on graphs have been recently generalized to hypergraphs (see, e.g. [10,11]). Several generalizations, e.g.…”

“…It is also known that hypertrees, hypercycles as well as graphs obtained recursively by specific operation of identification of appropriate vertices of some hypergraphs are sc-greedy (for more details see [10,11] and Section 2).…”

“…This section is based mainly on papers [10,11] and collects several results in the area of the hypergraph operations that preserve sc-greediness, and is intended to provide the necessary background for new operations on sc-greedy hypergraphs. We start with a union of hypergraphs.…”