arXiv:0704.2146v28 [math.CO] 19 Jul 2015 On C -homogeneous graphs and ordered pencils Italo J. Dejter University of Puerto Rico Rio Piedras, PR 00936-8377 italo.dejter@gmail.com Abstract The 42 ordered pencils of the Fano plane are the vertices of a con- nected 12-regular {K4,K2,2,2}-ultrahomogeneous graph G 1 3 . By nat- ural generalization, G 1 3 takes to connected graphs G σ r (2 <r ∈ Z and σ ∈ (0,r - 1) ∩ Z) fitting a definition of C-homogeneous graph G, with C being a class of graphs, that generalizes that of C-ultra- homogeneous graph by taking each induced subgraph Y of G in C with a distinguished fixed arc. In our case, C = {K2s ,Tts,t }, where t =2 σ+1 - 1, s =2 r-σ-1 and each edge of G shared by exactly one copy of the complete graph K2s and one of the Tur´an graph Tts,t . Moreover, if r - σ = 2, then G σ r is K4-ultrahomogeneous with order (2 r - 1)(2 r - 2), and 2 σ+1 edge-disjoint copies of K4 at each vertex. 1 C -homogeneous graphs A finite, undirected, simple graph G is homogeneous (resp. ultrahomoge- neous) if, whenever two induced subgraphs Y 1 = G[W 1 ],Y 2 = G[W 2 ] of respective vertex subsets W 1 ,W 2 of G are isomorphic, then some isomor- phism (resp. every isomorphism) of Y 1 onto Y 2 extends to an automorphism of G. Gardiner [5], Gol’fand and Klin [6, 8] and Reichard [9] gave explicit char- acterizations of ultrahomogeneous graphs. Let C be a class of graphs. Isaksen et al. [7] defined a graph G to be C - ultrahomogeneous (or C -UH)) if every isomorphism between induced sub- graphs of G in C extends to an automorphism of G. The 42 ordered pencils of the Fano plane are the vertices of a connected 12-regular C -UH graph G 1 3 , where C = {K 4 ,K 2,2,2 } [4]. 1