Combinatorially homogeneous graphs

Gardiner classified ultrahomogeneous graphs and posed the problem of defining “combinatorial homogeneity”. Later, Ronse proved that homogeneous graphs are ultrahomogeneous by classifying such graphs. In this paper, we give a direct proof that (suitably defined) combinatorially homogeneous graphs are ultrahomogeneous. Also, we clasify combinatorially C-homogeneous graphs.