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Finite 3-connected-set-homogeneous locally 2_n graphs and s-arc-transitive graphs
[article]
2022
arXiv
pre-print
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Furthermore, we give a good description of finite 3-connected-set-homogeneous but not 3-connected-homogeneous graphs which are locally 2_n and have solvable automorphism groups. ...
In this paper, we complete the classification of finite 3-connected-set-homogeneous graphs which are locally 2_n with n≥ 2, and all such graphs are line graphs of some specific 2-arc-transitive graphs. ...
connected-set-homogeneous graphs of girth 3 which are not 3-connected-homogeneous. ...
arXiv:2211.05888v1
fatcat:ame2gx3nmvfvtfsfhjepqrv77i
Homomorphism-homogeneous graphs
2010
Journal of Graph Theory
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Further we show that there are homomorphism-homogeneous graphs that do not contain the Rado graph as a spanning subgraph answering the second open question. ...
In particular we show, by giving a characterization of these graphs, that extendability to monomorphism or to homomorphism leads to the same class of graphs when defining homomorphism-homogeneity. ...
Gardiner [3] completed the classification of finite ultra-homogeneous graphs commenced by Sheehan [9] . Ronse [8] then showed that the finite ultra-homogeneous and homogeneous graphs coincide. ...
doi:10.1002/jgt.20478
fatcat:jq2sdupvl5e4bgdcppwsbdrdva
The independence number of HH-homogeneous graphs and a classification of MB-homogeneous graphs
[article]
2019
arXiv
pre-print
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We show that the independence number of a countably infinite HH-homogeneous graph that does not contain the Rado graph as a spanning subgraph is finite and present a classification of MB-homogeneous graphs ...
result in Section 3 to classify MB-homogeneous graphs. ...
Take S ∈ I σ(G) and a copy C 1 of K 3 in K S (we can find C 1 because K S = N (S) is an infinite HH-homogeneous graph with finite independence number). ...
arXiv:1902.03126v2
fatcat:a5xvt4rxmngapa3ppt7gdvfriu
IB-homogeneous graphs
[article]
2021
arXiv
pre-print
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We extend those results in this paper, showing that every IB-homogeneous graph is either ultrahomogeneous or MB-homogeneous, and thus all the IB-homogeneous graphs are known up to bimorphism-equivalence ...
The Lachlan-Woodrow Theorem identifies ultrahomogeneous graphs up to isomorphism. Recently, the present author and D. Hartman classified MB-homogeneous graphs up to bimorphism-equivalence. ...
One of them is the universal homogeneous trianglefree graph H 3 , which is the Fraïssé limit of all finite triangle-free graphs. ...
arXiv:1909.02920v2
fatcat:xqu3ndmnwzachchgq3busgg3ge
The Random Graph Revisited
[chapter]
2001
European Congress of Mathematics
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The random graph, or Rado's graph (the unique countable homogeneous graph) has made an appearance in many parts of mathematics since its first occurrences in the early 1960s. ...
Theorem 3. 3 . 3 For r ≥ 4, H r is not a normal Cayley graph of any countable group. ...
We call a graph with this property homogeneous; we have shown that a graph with the I-property is homogeneous. ...
doi:10.1007/978-3-0348-8268-2_15
fatcat:yw63li3lcnbazkshwm5oyj62iy
A finiteness theorem for primal extensions
2005
Discrete Mathematics
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If every homogeneous set of G has exactly two vertices then Ext(G) is a finite set. We extend this result to a wider class of graphs. ...
A graph is simple if it is isomorphic to an induced subgraph of the path P 4 . Theorem. If every homogeneous set of G induces a simple graph then Ext(G) is a finite set. ...
In particular, it was proposed to find necessary and sufficient conditions for a graph to have a finite number minimal primal extensions. ...
doi:10.1016/j.disc.2005.01.002
fatcat:7kb27af5kjdtfgs7dvbvv74abe
Homomorphism-homogeneous L-colored graphs
[article]
2012
arXiv
pre-print
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A full classification of finite MH-homogeneous L-colored graphs where L is a chain is provided, and we show that the classes MH and HH coincide. ...
In this paper we consider L-colored graphs, that is, undirected graphs without loops where sets of colors selected from L are assigned to vertices and edges. ...
Lemma 3. 2 . 2 Let G be a finite M H-homogeneous L-colored graph. ...
arXiv:1204.5879v1
fatcat:ds2ucyhmg5dzhord47zju4yupa
A census of infinite distance-transitive graphs
1998
Discrete Mathematics
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This paper describes some classes of infinite distance-transitive graphs. It has no pretensions to give a complete list, but concentrates on graphs which have no finite analogues. ...
A finite graph is 2-homogeneous if and only if its automorphism group is a rank 3 permutation group of even order. ...
These authors showed, moreover, that if a countable graph is 8-set homogeneous (this means that the above condition holds for subgraphs A and B with at most 8 vertices) but not 3-homogeneous, then it is ...
doi:10.1016/s0012-365x(98)00063-6
fatcat:tac5nwiyrnh4xboixwqzyk2hga
Infinite homogeneous bipartite graphs with unequal sides
1996
Discrete Mathematics
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We call a bipartite graph homogeneous if every finite partial automorphism which respects left and right can be extended to a total automorphism. ...
We show that there is a homogeneous (No, 2 ~°) bipartite graph of girth 4 (thus answering negatively a question by Kupitz and Perles), and that depending on the underlying set theory all homogeneous (No ...
All countable homogeneous graphs were classified in [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] , and countable tournaments were classified in 1, 9] (see also 1, 3] ). ...
doi:10.1016/0012-365x(94)00339-k
fatcat:xqoh6sogqbdzngdw7jwsij2yta
The classification of homomorphism homogeneous oriented graphs
[article]
2024
arXiv
pre-print
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An oriented graph is called homomorphism homogeneous if every homomorphism between finite induced subgraphs extends to an endomorphism. ...
In this paper we present a complete classification of the countable homomorphism homogeneous oriented graphs. ...
has a maximal lower bound or a minimal upper bound • no 4 -set has a midpoint, (6) an extension of the countable universal homogeneous strict poset, (7) an oriented graph isomorphic to 3 [ ], (2) ...
arXiv:2403.04393v1
fatcat:av2osljjgnbd7p6swz7moqzsje
The poset of morphism-extension classes of countable graphs
[article]
2020
arXiv
pre-print
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In this paper, we establish all equalities and inequalities between morphism-extension classes of countable (undirected, loopless) graphs. ...
Let XY_L,T denote the class of countably infinite L-structures that satisfy the axioms T and in which all homomorphisms of type X (these could be homomorphisms, monomorphisms, or isomorphisms) between finite ...
The infinite independent set is MI-homogeneous but not HI-homogeneous. 2. The universal homogeneous K n -free graph is II-homogeneous but not MIhomogeneous (Observation 11). 3. ...
arXiv:2002.07099v1
fatcat:w6npgsznn5gm7m2rwmxbc3bzle
Homogeneous graphs and stability
1976
Journal of the Australian Mathematical Society
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Let F be a graph with finite vertex set V. ...
We study finite, undirected, loopless graphs r = (V,E), with vertex set V = VT, edge set E C V x V, and automorphism group Autr = G. ...
The following conditions on a finite graph F are equivalent: (a) F is homogeneous, (b) F is locally homogeneous, (c) r is one ofthe graphs t • K r , t g \,r g 1; K,, r , t g 1, r g 1; C 5 ; L ( K 3 3 ...
doi:10.1017/s144678870001867x
fatcat:55bfdzg7iban3gdey6nqv67nui
Countable connected-homogeneous graphs
2010
Journal of combinatorial theory. Series B (Print)
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of countably many copies of a finite connected-homogeneous graph. ...
A graph is connected-homogeneous if any isomorphism between finite connected induced subgraphs extends to an automorphism of the graph. ...
The problem of classifying the countable set-homogeneous graphs is open, but in [5] the countable set-homogeneous graphs that are not 3-homogeneous are described (in fact, up to complementation there ...
doi:10.1016/j.jctb.2009.04.002
fatcat:unuaac5ywzeazdmf32cd3yqwjy
Ramsey Theory on Trees and Applications
[chapter]
2016
Lecture Notes in Computer Science
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In this talk, we concentrate on Ramsey theory on trees and applications to homogeneous structures. ...
For each positive integer k and each finite coloring of all k-sized subsets of the natural numbers, N, there is an infinite set M of natural numbers such that each k-sized subset of M has the same color ...
The universal homogeneous triangle-free graph is the Fraïssé limit of the Fraïssé class of all finite triangle-free graphs, which we shall denote by H 3 . ...
doi:10.1007/978-3-662-54069-5_2
fatcat:rwy5fsm2ovcvxjrischtx5nhlu
Infinite homogeneous bipartite graphs with unequal sides
[article]
1994
arXiv
pre-print
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on the underlying set theory all homogeneous (_0,_1) bipartite graphs may be isomorphic, or there may be 2^_1 many isomorphism types of (_0,_1) homogeneous graphs. ...
We call a bipartite graph homogeneous if every finite partial automorphism which respects left and right can be extended to a total automorphism. ...
If Γ = L, R, E , the complement graph of Γ, is the graph whose edge set is L × R \ E (3) If Γ is a bipartite graph and v ∈ Γ, the set Γ(v) = {u : u ∈ Γ, {v, u} ∈ E} is called the set of neighbors of v. ...
arXiv:math/9409204v1
fatcat:kup3ufngjnccbg7qidru25zgry
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