The minimum number of vertices for a triangle-free graph with image is 11 Original Research Article

It is well-known that the minimum number of vertices for a triangle-free 4-chromatic graph is 11, and the Grötzsch graph is just such a graph. In this paper, we show that every non-bipartite triangle-free graph image of order not greater than 10 has image. Combined with a known result by Hanson et al. [D. Hanson, G. MacGillivray, B. Toft, Choosability of bipartite graphs, Ars Combin. 44 (1996) 183–192] that every bipartite graph of order not greater than 13 is 3-choosable, we conclude that the minimum number of vertices for a triangle-free graph with image is also 11.