Two Chromatic Conjectures: One for Vertices and One for Edges

Abstract

Erdős, Faber, and Lovász conjectured that a pairwise edge-disjoint union of n copies of the complete graph K _n has chromatic number n. This seeming parlour puzzle has eluded proof for more than four decades, despite the attack by a few of this era’s more powerful combinatorial minds. Regarding edges, the list-colouring conjecture asserts, loosely, that list colouring is no more difficult than ordinary edge colouring. Probably first proposed by Vizing, this notorious conjecture—also having garnered the attention of leading combinatorialists—has itself defied proof for forty years. Like any good mature conjecture, both of these have spawned interesting mathematics vainly threatening their resolution. This chapter considers some of the related partial results in concert with the conjectures themselves.

This work was partially supported by a grant from the Simons Foundation (#279367 to Mark Kayll).

Appendix: Notation and Terminology

Our purpose here is to fix the notation and terminology appearing in this chapter and not to provide an exhaustive definition list for related nomenclature. Omissions of a combinatorial or graph-theoretic nature may be found in [10], while linear programming omissions can be rectified using [20]. Sets We denote the sets of real and nonnegative real numbers by $\mathbb{R}$ and $\mathbb{R}^{+}$, respectively. If n is a positive integer, then [n] means the set {1, 2, …, n}. If S is a set and k a nonnegative integer, then $\binom{S}{k}$ denotes the set of all k-element subsets of S.

Hypergraphs and Graphs A hypergraph consists of a finite set V of vertices, together with a finite multiset $\mathcal{H}$ of subsets of V; elements of $\mathcal{H}$ are called edges. We follow a common practice and use $\mathcal{H}$ to refer both to a hypergraph and its edge set. The order of a hypergraph is the cardinality of V and is usually denoted by n, while m is reserved for the size $\vert \mathcal{H}\vert$ of $\mathcal{H}$. Most hypergraphs here are simple, meaning they contain no singleton edges and any two distinct edges intersect in at most one vertex. The degree of a vertex is the number of edges containing it, and $\Delta (\mathcal{H})$ denotes the maximum degree in $\mathcal{H}$. A hypergraph is regular if every vertex has degree $\Delta$ and in this event is called Δ-regular. (We generally omit the argument from a hypergraph invariant when there’s no danger of ambiguity.) A hypergraph $\mathcal{H}$ is uniform if every edge $A \in \mathcal{H}$ contains the same number r of vertices and in this event is called r-uniform. We call $\mathcal{H}$ intersecting when $A \cap B\neq \varnothing$ for every pair A, B of edges in $\mathcal{H}$. One natural example of a hypergraph enjoying all of these last three properties is a projective plane $\mathcal{P}$ of ‘order’ q (for an integer q ≥ 2). Of course, as a hypergraph, such a $\mathcal{P}$ has order and size q ² + q + 1 and is (q + 1)-regular and (q + 1)-uniform, and pairwise edge intersections are all singletons. A degenerate projective plane (sometimes called a ‘near pencil’ in the literature) is a hypergraph on the vertex set [n] with edge set {{1, n}, {2, n}, …, {n − 1, n}, {1, 2, …, n − 1}}.

A multigraph G = (V, E) is a 2-uniform hypergraph with vertex set V and edge set E; this definition, though not quite standard, conveniently disallows G to contain loops, which get in the way of careful definitions of both vertex and edge colouring. A simple graph is a multigraph that is simple in the hypergraph sense. We sometimes use the generic ‘graph’ when there is no reason to be specific regarding multigraph or simple graph. The maximum number of vertices in a clique of G is denoted by ω(G).

Colouring For a graph G = (V, E) and a positive integer k, a k-colouring of G is a function σ: V → [k] such that

$$\displaystyle{ \sigma (x)\neq \sigma (y)\ \text{ whenever }\ \{x,y\} \in E. }$$

(11.15)

Such functions are usually called ‘proper colourings’, but we never consider improper colourings and thus dispense with the adjective. The least k for which G admits a k-colouring is G’s chromatic number χ(G). A k-list assignment L is a function that assigns to each vertex x of G a k-set (or ‘list’) L _x (of natural numbers, say). Given such an L, an L-colouring of G is a function $\sigma: V \rightarrow \bigcup _{x\in V }L_{x}$ such that σ(x) ∈ L _x for each x ∈ V and the usual colouring condition (11.15) is satisfied. The list-chromatic number $\chi _{L}(G)$ is the least integer k for which G admits an L-colouring for every k-list assignment L. Because one such L has each L _x  = [k], we always have $\chi _{L} \geq \chi$.

Both of χ, $\chi _{L}$ have edge and ‘total’ analogues. The chromatic index χ′(G) of G can be defined as the chromatic number of the line graph of G and likewise for the list-chromatic index $\chi _{L}'(G)$. The total graph of G = (V, E) has vertex set V ∪ E and an edge joining every pair of its vertices corresponding to an incident or adjacent pair of objects (vertices or edges) in G. The total chromatic number χ″(G) is the chromatic number of the total graph of G, while the total list-chromatic number $\chi ''_{L}(G)$ is defined analogously to $\chi _{L}'(G)$. It’s an exercise to prove that these invariants always satisfy

$$\displaystyle{ (\Delta + 1 \leq )\,\,\chi '' \leq \chi ''_{L} \leq \chi _{L}' + 2\,\,(\leq 2\Delta + 1). }$$

(11.16)

When we consider χ′ for hypergraphs $\mathcal{H}$, it’s useful to have in mind the connection with matchings. A matching in $\mathcal{H}$ is a set of pairwise disjoint edges of $\mathcal{H}$, and we write $\mathcal{M}$ for the set of matchings of $\mathcal{H}$. We denote by $\nu (\mathcal{H})$ the maximum size of a matching in $\mathcal{H}$, i.e., $\max \{\vert M\vert: M \in \mathcal{M}\}$. Now $\chi '(\mathcal{H})$ is the least size of a subset of $\mathcal{M}$ whose union is $\mathcal{H}$. This formulation may be cast in linear programming terms. First we define the fractional chromatic index $\chi '^{{\ast}}(\mathcal{H})$ as the optimal value of the LP (in the nonnegative orthant of $\mathbb{R}^{\mathcal{M}}$):

$$\displaystyle{ \begin{array}{rrcll} \min & \sum _{M\in \mathcal{M}}x(M) \\ \text{subject to}&\sum _{A\in M\in \mathcal{M}}x(M)& \geq &1&\text{for each }\ A \in \mathcal{H}.\\ \end{array} }$$

(11.17)

Notice that any optimal solution $x \in \mathbb{R}^{\mathcal{M}}$ to the LP (11.17), under the extra constraint that x have integer entries, must have {0, 1}-entries. Thus $\chi '(\mathcal{H})$ is the optimal value of this integer LP, whose linear relaxation (11.17) defines $\chi '^{{\ast}}(\mathcal{H})$. We also have one occasion to refer to the LP dual of problem (11.17) (in the nonnegative orthant of $\mathbb{R}^{\mathcal{H}}$):

$$\displaystyle{ \begin{array}{rrcll} \max & \sum _{A\in \mathcal{H}}w(A) \\ \text{subject to}&\sum _{A\in M}w(A)& \leq &1&\text{for each }\ M \in \mathcal{M}.\\ \end{array} }$$

(11.18)

In (11.8)—see Sect. 11.1.1—it would have been natural to write $\chi _{L}'^{{\ast}}$ in place of χ′^∗, and indeed, we could have done so because these two invariants turn out to be the same; see, e.g., [55].