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Forbidden cycles in metrically homogeneous graphs
[article]

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*. ...

##
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Combinatorial Properties of Metrically Homogeneous Graphs
[article]

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

arXiv:1805.07425v1
fatcat:jia72p5m3vc6hflolgkkeqylvi
*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. ...##
###
Completing graphs to metric spaces
[article]

2019
*
arXiv
*
pre-print

We prove that certain classes of

arXiv:1706.00295v3
fatcat:sfxputmtkve5xeg26qtkf5jwl4
*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 ...##
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Ramsey expansions of metrically homogeneous graphs
[article]

2017
*
arXiv
*
pre-print

At the heart of our proof is a canonical way of completing edge-labelled

arXiv:1707.02612v2
fatcat:b5qkd5fhijdxnoiwlus4zlkztm
*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*. ...##
###
Twists and Twistability
[article]

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 < ∞. ...

##
###
Extending partial isometries of antipodal graphs
[article]

2019
*
arXiv
*
pre-print

We prove EPPA (extension property for partial automorphisms) for all antipodal classes from Cherlin's list of

arXiv:1901.04426v2
fatcat:gmjf4l76lze3ddyw4pnemaauai
*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. ...##
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A census of infinite distance-transitive graphs

1998
*
Discrete Mathematics
*

This paper describes some classes of infinite distance-transitive

doi:10.1016/s0012-365x(98)00063-6
fatcat:tac5nwiyrnh4xboixwqzyk2hga
*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. ...##
###
Completely separable graphs

1990
*
Discrete Applied Mathematics
*

"Completely separable

doi:10.1016/0166-218x(90)90131-u
fatcat:ax34bnejtrclba6g44aozwjvpq
*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*. ...##
###
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. ...##
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Semigroup-valued metric spaces
[article]

2019
*
arXiv
*
pre-print

Sauer's S-

arXiv:1810.08963v2
fatcat:5pta7k7sabfo3feytw7mop5pye
*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. ...##
###
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) ...

##
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Homomorphism and embedding universal structures for restricted classes
[article]

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] . ...

##
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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

1995
*
Mathematical programming
*

Given a

doi:10.1007/bf01592243
fatcat:duc33ngdijddjnf6pzoj66ttcm
*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. ...##
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Infinite Limits of Finite-Dimensional Permutation Structures, and their Automorphism Groups: Between Model Theory and Combinatorics
[article]

2018
*
arXiv
*
pre-print

As a first approximation to this problem, we prove the undecidability of the corresponding decision problem

arXiv:1805.04219v1
fatcat:w3vpswxpu5gx7htiuead545yg4
*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. ...
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