1,723 research outputs found

    Infinite primitive and distance transitive directed graphs of finite out-valency

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    We give certain properties which are satisfied by the descendant set of a vertex in an infinite, primitive, distance transitive digraph of finite out-valency and provide a strong structure theory for digraphs satisfying these properties. In particular, we show that there are only countably many possibilities for the isomorphism type of such a descendant set, thereby confirming a conjecture of the first Author. As a partial converse, we show that certain related conditions on a countable digraph are sufficient for it to occur as the descendant set of a primitive, distance transitive digraph

    On the existence and number of (k+1)(k+1)-kings in kk-quasi-transitive digraphs

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    Let D=(V(D),A(D))D=(V(D), A(D)) be a digraph and k≥2k \ge 2 an integer. We say that DD is kk-quasi-transitive if for every directed path (v0,v1,...,vk)(v_0, v_1,..., v_k) in DD, then (v0,vk)∈A(D)(v_0, v_k) \in A(D) or (vk,v0)∈A(D)(v_k, v_0) \in A(D). Clearly, a 2-quasi-transitive digraph is a quasi-transitive digraph in the usual sense. Bang-Jensen and Gutin proved that a quasi-transitive digraph DD has a 3-king if and only if DD has a unique initial strong component and, if DD has a 3-king and the unique initial strong component of DD has at least three vertices, then DD has at least three 3-kings. In this paper we prove the following generalization: A kk-quasi-transitive digraph DD has a (k+1)(k+1)-king if and only if DD has a unique initial strong component, and if DD has a (k+1)(k+1)-king then, either all the vertices of the unique initial strong components are (k+1)(k+1)-kings or the number of (k+1)(k+1)-kings in DD is at least (k+2)(k+2).Comment: 17 page

    Countable connected-homogeneous digraphs

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    A digraph is connected-homogeneous if every isomorphism between two finite connected induced subdigraphs extends to an automorphism of the whole digraph. In this paper, we completely classify the countable connected-homogeneous digraphs.Comment: 49 page

    Constructing continuum many countable, primitive, unbalanced digraphs

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    AbstractWe construct continuum many non-isomorphic countable digraphs which are highly arc transitive, have finite out-valency and infinite in-valency, and whose automorphism groups are primitive

    k-colored kernels

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    We study kk-colored kernels in mm-colored digraphs. An mm-colored digraph DD has kk-colored kernel if there exists a subset KK of its vertices such that (i) from every vertex v∉Kv\notin K there exists an at most kk-colored directed path from vv to a vertex of KK and (ii) for every u,v∈Ku,v\in K there does not exist an at most kk-colored directed path between them. In this paper, we prove that for every integer k≥2k\geq 2 there exists a (k+1)% (k+1)-colored digraph DD without kk-colored kernel and if every directed cycle of an mm-colored digraph is monochromatic, then it has a kk-colored kernel for every positive integer k.k. We obtain the following results for some generalizations of tournaments: (i) mm-colored quasi-transitive and 3-quasi-transitive digraphs have a kk% -colored kernel for every k≥3k\geq 3 and k≥4,k\geq 4, respectively (we conjecture that every mm-colored ll-quasi-transitive digraph has a kk% -colored kernel for every k≥l+1)k\geq l+1), and (ii) mm-colored locally in-tournament (out-tournament, respectively) digraphs have a kk-colored kernel provided that every arc belongs to a directed cycle and every directed cycle is at most kk-colored
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