arXiv:0704.2146v19 [math.CO] 13 Feb 2012 Constructing {K 2s ,T ts,t }-Homogeneous Graphs via Ordered Pencils Italo J. Dejter University of Puerto Rico Rio Piedras, PR 00936-8377 ijdejter@uprrp.edu Abstract Let C be a class of graphs. A restricted concept of C-homogeneous graph that generalizes the concept of C- ultrahomogeneous graph [5] is given. We verify that a construction that employs ordered pencils of binary projective spaces to yield non-line-graphical {K2s ,Tts,t }-homogeneous graphs, including the {K4,K2,2,2}- ultrahomogeneous graph of [2], yields 20 such graphs that are not {K2s ,Tts,t }-ultrahomogeneous, but 5 of them still are {K4}-ultrahomogeneous. It is conjectured that this produces an infinite number of graphs with similar properties, which are translatable into point-line configuration terms. 1 Introduction Let G be a finite, undirected, simple graph, and let W 1 ,W 2 be vertex subsets of G. In Gardiner’s paper [3], G is said to be homogeneous (resp. ultrahomogeneous) if whenever the induced subgraphs G[W 1 ] and G[W 2 ] are isomorphic, some isomorphism (resp. every isomorphism) of G[W 1 ] onto G[W 2 ] extends to an automorphism of G. Gardiner [3] gave an explicit characterization of ultrahomogeneous graphs, (independently worked out by Gol’fand and Klin [4]). In [6], Ronse showed that a homogeneous graph is ultrahomogeneous. Let C be a class of graphs. In [5], Isaksen et al. define a graph G to be C -ultrahomogeneous if every isomorphism between induced subgraphs belonging to C extends to an automorphism of G. There is an absence of a notion of C -homogeneous graphs comparable to that of the C -ultrahomogeneous graphs of [5] in the sense of Ronse’s work [6]. The following restricted definition of C -homogeneity between isomorphic induced components of graphs in C , restricted in the sense that they are anchored at specific arcs, is proposed to fulfill this absence, for it allows then the presentation of non-line-graphical restricted C -homogeneous graphs that are not C -ultrahomogeneous, as is shown for 20 cases, but conjectured in more generality, below. Definition 1.1 A graph G is C -homogeneous if, for any two isomorphic induced subgraphs Y 1 ,Y 2 ∈C in G and any fixed arcs (v i ,w i ) of Y i , with i =1, 2, there exists an isomorphism f : Y 1 → Y 2 extending to an automorphism of G and such that f (v 1 )= v 2 and f (w 1 )= w 2 , so that f (v 1 ,w 1 )=(v 2 ,w 2 ). In Definition 1.1, if Y 1 = Y 2 then Y 1 (and hence any graph in C induced by G) must be 1-arc transitive. If C is a minimal class containing two nonisomorphic graphs X 1 and X 2 , then a C -homogeneous graph is said to be {X 2 ,X 1 }-homogeneous. Clearly, any C -ultrahomogeneous graph is C -homogeneous. Let (r,σ) ∈ Z 2 , r> 2 and σ ∈ (0,r − 1). In addition, let t =2 σ+1 − 1 and s =2 r−σ−1 . Now, let K 2s be the complete graph on 2s vertices and let T ts,t be the t-partite (Tur´ an) graph on s vertices per part (a total of ts vertices). A construction of graphs G σ r below allows to pose the following conjecture, with G 1 3 seen to be {K 4 ,K 2,2,2 }-ultrahomogeneous already in [2]. Conjecture 1.2 There exists a connected non-line-graphical {K 2s ,T ts,t }-homogeneous graph G σ r that is not {K 2s ,T ts,t }-ultrahomogeneous for r> 3. 1