Finite 33-connected homogeneous graphs

A finite graph \G is said to be {\em $(G,3)$-$($connected$)$ homogeneous} if every isomorphism between any two isomorphic (connected) subgraphs of order at most $3$ extends to an automorphism $g\in G$ of the graph, where $G$ is a group of automorphisms of the graph. In 1985, Cameron and Macpherson determined all finite $(G, 3)$-homogeneous graphs. In this paper, we develop a method for characterising $(G,3)$-connected homogeneous graphs. It is shown that for a finite $(G,3)$-connected homogeneous graph \G=(V, E), either G_v^{\G(v)} is $2$--transitive or G_v^{\G(v)} is of rank $3$ and \G has girth $3$, and that the class of finite $(G,3)$-connected homogeneous graphs is closed under taking normal quotients. This leads us to study graphs where $G$ is quasiprimitive on $V$. We determine the possible quasiprimitive types for $G$ in this case and give new constructions of examples for some possible types

Further Results on Homogeneous Two-Weight Codes

The results of [1,2] on linear homogeneous two-weight codes over finite Frobenius rings are exended in two ways: It is shown that certain non-projective two-weight codes give rise to strongly regular graphs in the way described in [1,2]. Secondly, these codes are used to define a dual two-weight code and strongly regular graph similar to the classical case of projective linear two-weight codes over finite fields [3].Comment: 7 pages, reprinted from the conference proceedings of the Fifth International Workshop on Optimal Codes and Related Topics (OC2007

Bounds for the annealed return probability on large finite percolation clusters

Bounds for the expected return probability of the delayed random walk on finite clusters of an invariant percolation on transitive unimodular graphs are derived. They are particularly suited for the case of critical Bernoulli percolation and the associated heavy-tailed cluster size distributions. The upper bound relies on the fact that cartesian products of finite graphs with cycles of a certain minimal size are Hamiltonian. For critical Bernoulli bond percolation on the homogeneous tree this bound is sharp. The asymptotic type of the expected return probability for large times t in this case is of order of the 3/4'th power of 1/t.Comment: New result for the particular case of homogeneous trees illustrates sharpness of the boun

Forbidden cycles in metrically homogeneous graphs

Aranda, Bradley-Williams, Hubi\v{c}ka, Karamanlis, Kompatscher, Kone\v{c}n\'y and Pawliuk recently proved that for every primitive 3-constrained space $\Gamma$ of finite diameter $\delta$ from Cherlin's catalogue of metrically homogeneous graphs there is a finite family $\mathcal F$ of $\{1,2,\ldots, \delta\}$-edge-labelled cycles such that each $\{1,2,\ldots, \delta\}$-edge-labelled graph is a (not necessarily induced) subgraph of $\Gamma$ if and only if it contains no homomorphic images of cycles from $\mathcal F$. This analysis is a key to showing that the ages of metrically homogeneous graphs have Ramsey expansions and the extension property for partial automorphisms. In this paper we give an explicit description of the cycles in families $\mathcal F$. This has further applications, for example, interpreting the graphs as semigroup-valued metric spaces or homogenizations of $\omega$-categorical $\{1,\delta\}$-edge-labelled graphs.Comment: 24 pages, 2 table

Relational Width of First-Order Expansions of Homogeneous Graphs with Bounded Strict Width

Solving the algebraic dichotomy conjecture for constraint satisfaction problems over structures first-order definable in countably infinite finitely bounded homogeneous structures requires understanding the applicability of local-consistency methods in this setting. We study the amount of consistency (measured by relational width) needed to solve CSP(?) for first-order expansions ? of countably infinite homogeneous graphs ? := (A; E), which happen all to be finitely bounded. We study our problem for structures ? that additionally have bounded strict width, i.e., for which establishing local consistency of an instance of CSP(?) not only decides if there is a solution but also ensures that every solution may be obtained from a locally consistent instance by greedily assigning values to variables, without backtracking. Our main result is that the structures ? under consideration have relational width exactly (2, ?_?) where ?_? is the maximal size of a forbidden subgraph of ?, but not smaller than 3. It beats the upper bound: (2 m, 3 m) where m = max(arity(?)+1, ?, 3) and arity(?) is the largest arity of a relation in ?, which follows from a sufficient condition implying bounded relational width given in [Manuel Bodirsky and Antoine Mottet, 2018]. Since ?_? may be arbitrarily large, our result contrasts the collapse of the relational bounded width hierarchy for finite structures ?, whose relational width, if finite, is always at most (2,3)