A note on uncountable chordal graphs

We show that if X is a chordal graph containing no clique of size μ (μ an infinite cardinal) then the chromatic (even coloring) number of X is at most μ. The same conclusion holds if the condition ‘is chordal’ is replaced by ‘contains no induced C4 (or Kk,k for k finite)’. In [9] Wagon asked if the following holds. If X is a chordal graph with Kω 6≤ X then Chr(X) ≤ ω. This was proved by Halin in [5]. Here we improve this result. We show that if μ is an infinite cardinal, X is a chordal graph such that Kμ 6≤ X, then Chr(X) ≤ μ, in fact even the coloring number of X is at most μ. With a different argument we show the same result if only an induced C4 (or any induced Kk,k) is excluded. An example of Galvin’s shows that for every infinite cardinal μ there is an interval graph X containing no Kμ+ but Chr(X) = μ , that is, X is not perfect. Notation. Definitions. We use the notation and definitions of axiomatic set theory. In particular, ordinals are von Neumann ordinals, and each cardinal is identified with the least ordinal of that cardinality. For the notions of regular cardinals, closed unbounded and stationary sets, the reader can consult [4]. A graph is an arbitrary set of unordered pairs of some set V , the set of vertices. If v ∈ V , then N(v) is the set of neighbors of v, i.e., N(v) = {w : {v, w} ∈ X}. If κ is a cardinal, then Kκ is the complete graph on κ vertices. If a, b are cardinals, then Ka,b is the complete bipartite graph with bipartition classes of cardinality a, b, respectively. Cn denotes the circuit of length n. A graph is chordal if it does not contain an induced Cn for n ≥ 4.