The minimum number of vertices for a triangle-free graph with χl(G)=4χl(G)=4 is 11

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 GG of order not greater than 10 has χl(G)=3χl(G)=3. 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 χl(G)=4χl(G)=4 is also 11.