By Vadim Lozin
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A class of graphs is called χ-bounded if there is a function f such that for every graph G in this class, χ(G) ≤ f (ω(G)). 48 There are several interesting conjectures about χ-bounded classes. One of them deals with classes defined by a single forbidden induced subgraph G. A necessary condition for a class F ree(G) to be χ-bounded is that G must be acyclic, which follows from Theorem 115. Corollary 117. e. a graph without cycles). Proof. Assume G is not a forest, and Ck be a chordless cycle contained in G.
Vk be a longest path in G. By the maximality of P , all the neighbours of v0 and all the neighbours of vk lie on P . Hence for at least n/2 indices 0 ≤ i ≤ k − 1, we have vi vk ∈ E and for at least n/2 indices 1 ≤ i ≤ k, we have v0 vi ∈ E. The second part of the last sentence can be rephrased as follows: for at least n/2 indices 0 ≤ i ≤ k − 1, we have v0 vi+1 ∈ E. Since k < n, by the pigeon hole principle, there is an index 0 ≤ i ≤ k −1 such that both vi vk ∈ E and v0 vi+1 ∈ E. We claim that the cycle C = v0 vi+1 vi+2 .
E. an augmenting path. But this contradicts the fact that M is maximum. Let G = (A, B, E) be a bipartite graph. e. covering each vertex of A), then |N (S)| ≥ |S| for all S ⊆ A. (1) In other words, (1) is a necessary condition for the existence of a matching covering A. Philip Hall proved in 1935 that this condition is also sufficient. Theorem 91 (Hall, 1935). G contains a matching covering A if and only if (1) holds. Proof. In order to prove the theorem we assume that G satisfies condition (1) and find a matching covering A.