The Erdős Bipartification Conjecture Is True in the Special Case of Andrásfai Graphs

Let the Andrásfai graph Andk be defined as the graph with vertex set {v0, v1, . . . , v3k−2} and two vertices vi and vj being adjacent iff |i − j| ≡ 1 mod 3. The graphs Andk are maximal triangle-free and play a role in characterizing triangle-free graphs with large minimum degree as homomorphic preimages. A minimal bipartification of a graph G is defined as a set of edges F ⊂ E(G) having the property that the graph (V (G), E(G)\F ) is bipartite and for every e ∈ F the graph (V (G), E(G)\(F\e)) is not bipartite. In this note it is shown that there is a minimal bipartification Fk of Andk which consists of exactly j k 4 k edges. This equals j 1 36 ` |Andk|+ 1 ́ 2 k , where | · | denotes the number of vertices of a graph. For all k this is consistent with a conjecture of Paul Erdős that every triangle-free graph G can be made bipartite by deleting at most 1 25 |G| edges. Bipartifications like Fk may be useful for proving that arbitrary homomorphic preimages of an Andrásfai graph can be made bipartite by deleting at most 1 25 |G| edges.