A New Characterization of Trivially Perfect Graphs

A graph $G$ is \emph{trivially perfect} if for every induced subgraph the cardinality of the largest set of pairwise nonadjacent vertices (the stability number) $\alpha(G)$ equals the number of (maximal) cliques $m(G)$. We characterize the trivially perfect graphs in terms of vertex-coloring and we extend some definitions to infinite graphs.


Introduction
Let G be a finite graph. A coloring (vertex-coloring) of G with k colors is a surjective function that assigns to each vertex of G a number from the set {1, . . . , k}. A coloring of G is called pseudo-Grundy if each vertex is adjacent to some vertex of each smaller color. The pseudo-Grundy number γ(G) is the maximum k for which a pseudo-Grundy coloring of G exists (see [5,6]).
A coloring of G is called proper if any two adjacent vertices have different color. A proper pseudo-Grundy coloring of G is called Grundy. The Grundy number Γ(G) (also known as the first-fit chromatic number) is the maximum k for which a Grundy coloring of G exists (see [6,11]).
Since there must be α(G) distinct cliques containing the members of a maximum stable set, clearly, www.ejgta.org A new characterization of trivially perfect graphs | C. Rubio-Montiel where θ denotes the clique cover (the least number of cliques of G whose union covers V (G)), ω denotes the clique number and χ denotes the chromatic number. Let a, b ∈ {α, θ, m, ω, χ, Γ, γ} such that a = b. A graph G is called ab-perfect if for every induced subgraph H of G, a(H) = b(H). This definition extends the usual notion of perfect graph introduced by Berge [3], with this notation a perfect graph is denoted by ωχ-perfect. The concept of the ab-perfect graphs was introduced earlier by Christen and Selkow in [7] and extended in [17] and [1,2]. A graph G without an induced subgraph H is called H-free. A graph H 1 -free and H 2 -free is called (H 1 , H 2 )-free.
Some important known results are the following: Lóvasz proved in [13] that a graph G is ωχ-perfect if and only its complement is ωχ-perfect. Consequently, a graph G is ωχ-perfect if and only if G is αθ-perfect, see also [4,5,12]. By Equation (1), a graph αm-perfect is "trivially" perfect (see [9,10]). Chudnovsky, Robertson, Seymour and Thomas proved in [8] that a graph G is ωχ-perfect if and only if G and its complement are C 2k+1 -free for all k ≥ 2. Christen and Selkow proved in [7] that for any graph G the following are equivalent: G is ωΓ-perfect, G is χΓ-perfect, and G is P 4 -free.
The remainder of this paper is organized as follows: In Section 2: Characterizations are given of the families of finite graphs: (i) θm-perfect graphs, (ii) αm-perfect graphs (trivially perfect graphs), (iii) ωγ-perfect graphs and (iv) χγ-perfect graphs. In Section 3: We further extend some definitions to locally finite graphs and denumerable graphs.

Characterizations for finite graphs
There exist several trivially perfect graph characterizations, e.g. [2,9,14,15,16]. We will use the following equivalence to prove Theorem 2.2: A consequence of Theorem 2.1 is the following characterization of θm-perfect and trivially perfect graphs.
We now characterize the ωγ-perfect and χγ-perfect graphs. In the following result, one should note that the finiteness of G is not necessary for the proof, the finiteness of ω(G) is sufficient.
Theorem 2.2. For any graph G the following are equivalent: 1 G is (C 4 , P 4 )-free, 2 G is ωγ-perfect, and 3 G is χγ-perfect.
Proof. To prove 1 ⇒ 2 assume that G is (C 4 , P 4 )-free. Let ς be a pseudo-Grundy coloring of G with γ(G) colors. We will prove by induction on n that for n ≤ γ(G), G contains a complete subgraph of n vertices with the n highest colors of ς. This proves (for n = γ(G)) that G is ωγ-perfect since every induced subgraph of G is (C 4 , P 4 )-free.
For n = 1, there exists a vertex with color γ(G), then the assertion is trivial. Let us now suppose that we have n − 1 vertices v 1 , . . . , v n−1 in the n − 1 highest colors such that they are the vertices of a complete subgraph, and define V i as the set of vertices colored γ(G) − (n − 1) by ς adjacent to v i (1 ≤ i < n). Since ς is a pseudo-Grundy coloring, none V i is empty. Any two such sets are comparable with respect to inclusion, otherwise there must be vertices p in V i \ V j and q in V j \ V i and the subgraph induced by {p, v i , v j , q} would be isomorphic to C 4 or P 4 . Therefore the n − 1 sets V i are linearly ordered with respect to inclusion, and there is a k (1 ≤ k < n) with Thus there is a vertex v n in V k which is colored with γ(G) − n + 1 by ς and is adjacent to each of The proof of 2 ⇒ 3 is immediate from Equation (1). To prove 3 ⇒ 1 note that if H ∈ {C 4 , P 4 } then χ(H) = 2 and γ(H) = 3 hence the implication is true (see Fig 1).

Extensions for infinite graphs
We presuppose here the axiom of choice. The definitions of pseudo-Grundy coloring with n colors and of proper coloring with n colors of a finite graph are generalizable to any cardinal number. It is defined the chromatic number χ of a graph as the smallest cardinal κ such that the graph has a proper coloring with κ colors. The clique number ω of a graph as the supremum of the cardinalities of the complete subgraphs of the graph (see [7]). Similarly, for any ordinal number β (such that |β| = κ), a pseudo-Grundy coloring of a graph with κ colors is a coloring of the vertices of the graph with the elements of β such that for any β ′′ < β ′ and any vertex v colored β ′ there is a vertex colored β ′′ adjacent to v. The pseudo-Grundy number γ of a graph is the supremum of the cardinalities κ for which there is a pseudo-Grundy coloring of the graph with β such that |β| = κ.
Next we prove a generalization of Theorem 2.2 for some classes of infinite graphs. Afterwards we show that there exists a graph, not belonging to these classes, for which the theorem does not hold. Proof. To prove 1 ⇒ 2 , let H be an induced subgraph of G. If ω(H) is finite, we can use the proof of Theorem 2.2 to show that γ(H) = ω(H). In otherwise ω(H) is infinite, then γ(H) = ω(H), because γ(H) is at most the supremum of the degrees of the vertices of H, which is at most ℵ 0 , if G is locally finite or denumerable.
The implications 2 ⇒ 3 and 3 ⇒ 1 hold for any graph, finite or not.