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Forbidden cycles in metrically homogeneous graphs [article]

Jan Hubička, Michael Kompatscher, Matěj Konečný
2018 arXiv   pre-print
In this paper we give an explicit description of the cycles in families F.  ...  This has further applications, for example, interpreting the graphs as semigroup-valued metric spaces or homogenizations of ω-categorical {1,δ}-edge-labelled graphs.  ...  In this paper we give an alternative description of metrically homogeneous graphs by means of forbidden cycles.  ... 
arXiv:1808.05177v1 fatcat:axjmnjrntzabvj4w2h67u62y2q

Combinatorial Properties of Metrically Homogeneous Graphs [article]

Matěj Konečný
2018 arXiv   pre-print
This thesis is a contribution to the programme; we find Ramsey expansions of the primitive 3-constrained classes from Cherlin's catalogue of metrically homogeneous graphs.  ...  A key ingradient is an explicit combinatorial algorithm to fill-in the missing distances in edge-labelled graphs to obtain structures from Cherlin's classes.  ...  Introduction A metrically homogeneous graph is a countable graph which gives rise to a homogeneous metric space when one computes the distances between all vertices.  ... 
arXiv:1805.07425v1 fatcat:jia72p5m3vc6hflolgkkeqylvi

Completing graphs to metric spaces [article]

Andrés Aranda, David Bradley-Williams, Eng Keat Hng, Jan Hubička, Miltiadis Karamanlis, Michael Kompatscher, Matěj Konečný, Micheal Pawliuk
2019 arXiv   pre-print
We prove that certain classes of metrically homogeneous graphs omitting triangles of odd short perimeter as well as triangles of long perimeter have the extension property for partial automorphisms and  ...  The graph G is metric if and only if it does not contain a non-metric cycle, that is, an edge-labelled graph cycle such that one distance in the cycle is greater than the sum of the remaining distances  ...  do) or the Ramsey property when applied to slight modifications of the generalised algorithm in nearly all other classes of metrically homogeneous graphs with finite diameter in Cherlin's catalogue, as  ... 
arXiv:1706.00295v3 fatcat:sfxputmtkve5xeg26qtkf5jwl4

Ramsey expansions of metrically homogeneous graphs [article]

Andrés Aranda, David Bradley-Williams, Jan Hubička, Miltiadis Karamanlis, Michael Kompatscher, Matěj Konečný, Micheal Pawliuk
2017 arXiv   pre-print
At the heart of our proof is a canonical way of completing edge-labelled graphs to metric spaces in Cherlin's classes.  ...  We show that, with the exception of tree-like graphs, all metric spaces in the catalogue have precompact Ramsey expansions (or lifts) with the expansion property.  ...  The third author would like to thank to Gregory Cherlin for discussions concerning finite set of obstacles of metrically homogeneous graphs.  ... 
arXiv:1707.02612v2 fatcat:b5qkd5fhijdxnoiwlus4zlkztm

Twists and Twistability [article]

Rebecca Coulson
2018 arXiv   pre-print
Metrically homogeneous graphs are connected graphs which, when endowed with the path metric, are homogeneous as metric spaces.  ...  We find all permutations of the language which are associated with twisted automorphisms of metrically homogeneous graphs.  ...  A p-cycle embeds isometrically in Γ. 2. p ≤ 2δ + 1. 3. p = 2K 1 + 1. This fact implies the following. Corollary 5.14.1. Let Γ be a metrically homogeneous graph with K 1 < ∞.  ... 
arXiv:1802.00467v1 fatcat:olxu3up5krcn5jqswbmd6pzspq

Extending partial isometries of antipodal graphs [article]

Matěj Konečný
2019 arXiv   pre-print
We prove EPPA (extension property for partial automorphisms) for all antipodal classes from Cherlin's list of metrically homogeneous graphs, thereby answering a question of Aranda et al.  ...  It is done by combining the recent strengthening of the Herwig–Lascar theorem by Hubička, Nešetřil and the author with the ideas of the proof of EPPA for two-graphs by Evans et al.  ...  Finite cycles of size at least 6 are examples of metrically homogeneous graphs which are not homogeneous. Remark 2.1.  ... 
arXiv:1901.04426v2 fatcat:gmjf4l76lze3ddyw4pnemaauai

A census of infinite distance-transitive graphs

Peter J. Cameron
1998 Discrete Mathematics  
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.  ...  Homogeneity can be weakened in another way. Any connected graph can be regarded as a metric space.  ...  Homogeneous metric spaces In this section we construct distance-regular graphs of arbitrary diameter (possibly infinite). They are not homogeneous as graphs, so we construct them as metric spaces.  ... 
doi:10.1016/s0012-365x(98)00063-6 fatcat:tac5nwiyrnh4xboixwqzyk2hga

Completely separable graphs

Peter L. Hammer, Frédéric Maffray
1990 Discrete Applied Mathematics  
"Completely separable graphs" are then derived and characterized in particular by the existence of two crossing chords in any cycle of length at least five.  ...  This implies that completely separable graphs are perfect.  ...  Indeed the graphs shown in Fig. 1 (long chordless cycle, "house", "gem" and "domino") clearly contain no proper homogeneous set and no hinge. Furthermore they are minimal among nonseparable graphs.  ... 
doi:10.1016/0166-218x(90)90131-u fatcat:ax34bnejtrclba6g44aozwjvpq

Page 12 of Mathematical Reviews Vol. , Issue 88e [page]

1988 Mathematical Reviews  
For k > 3, let f(k) denote the largest integer j such that in every k-connected k-regular graph, every set of j points lies on some cycle.  ...  General estimates are given for sat(n;¥%) for some special families of forbidden graphs ¥ , and in some of these instances the structure of the minimal graphs is described. R. H.  ... 

Semigroup-valued metric spaces [article]

Matěj Konečný
2019 arXiv   pre-print
Sauer's S-metric spaces, Conant's generalised metric spaces, Braunfeld's Λ-ultrametric spaces or Cherlin's metrically homogeneous graphs).  ...  results on Λ-ultrametric spaces and the results of Aranda et al. on Cherlin's primitive 3-constrained metrically homogeneous graphs.  ...  Observe that in particular M ⊕ x = M for every x.  ... 
arXiv:1810.08963v2 fatcat:5pta7k7sabfo3feytw7mop5pye

Page 1681 of Mathematical Reviews Vol. , Issue 2000c [page]

2000 Mathematical Reviews  
In this paper, the authors establish the foundation for a theory of study of these graphs, and decide what graphs arise as an SIG(M, X) for a given metric space.  ...  (i = 1,2,---,r) with length s are elementary cycles in D(A); (2) if A contains d pos- itive diagonal elements, and N(A) is the number of 1’s in A, then y(A)>n+ [n/d|-2 if N(A)=n+d; y(A) > [2(n—1)/(N(A)  ... 

Homomorphism and embedding universal structures for restricted classes [article]

Jan Hubička, Jaroslav Nešetřil
2014 arXiv   pre-print
Forbidden cycles and Urysohn spaces (binary lifts) We turn briefly our attention to binary lifts.  ...  It is well known that universal and homogeneous metric space exists for several classes of metric spaces [10, 33] . Analogously we have: Theorem 5.2 There exists a generic even-odd metric space U.  ...  We shall also remark that the universal structures for F = {C l } can also be constructed as metrically homogeneous graphs, see [5] .  ... 
arXiv:0909.4939v2 fatcat:l7guirx6bzblnlipugy5amse2y

Page 761 of Mathematical Reviews Vol. 57, Issue 3 [page]

1979 Mathematical Reviews  
Liu, A generalization of Ramsey theory for graphs— with stars and complete graphs as forbidden subgraphs (pp. 155- 161); G. Berman and A. W.  ...  Hemminger, Line graphs, total graphs and forbidden subgraphs (pp. 345-353); Edward Howorka, A characterization of Ptolemaic graphs; survey of results (pp. 355-361); F. K.  ... 

One-third-integrality in the max-cut problem

Monique Laurent, Svatopluk Poljak
1995 Mathematical programming  
Given a graph G = (V, E), the metric polytope S(G) is defined by the inequalities x(F) -x(C \ F) ~< IF I -1 for F C C, IF[ odd, C cycle of G, and 0 ~< Xe ~< 1 for e C E.  ...  We prove that the class of 1/d-integral graphs is closed under minors, and we present several minimal forbidden minors for ½-integrality.  ...  Take two graphs Gi = (Vi, Ei), i = 1,2, having a common edge f and denote their 2-sum by G = (V,E). We suppose that G i is box ½-integral for i = 1,2, and we show that G is box ½-integral.  ... 
doi:10.1007/bf01592243 fatcat:duc33ngdijddjnf6pzoj66ttcm

Infinite Limits of Finite-Dimensional Permutation Structures, and their Automorphism Groups: Between Model Theory and Combinatorics [article]

Samuel Braunfeld
2018 arXiv   pre-print
As a first approximation to this problem, we prove the undecidability of the corresponding decision problem in the category of graphs.  ...  Modifying this proof also gives the undecidability, in the category of graphs, of the corresponding decision problem for the joint homomorphism property, which is of interest in infinite-domain constraint  ...  of homogeneous ordered graphs was undertaken.  ... 
arXiv:1805.04219v1 fatcat:w3vpswxpu5gx7htiuead545yg4
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