On the order of countable graphs

A set of graphs is said to be independent if there is no homomorphism between distinct graphs from the set. We consider the existence problems related to the independent sets of countable graphs. While the maximal size of an independent set of countable graphs is 2^omega the On Line problem of extending an independent set to a larger independent set is much harder. We prove here that singletons can be extended (``partnership theorem''). While this is the best possible in general, we give structural conditions which guarantee independent extensions of larger independent sets. This is related to universal graphs, rigid graphs and to the density problem for countable graphs

How proofs are prepared at Camelot

We study a design framework for robust, independently verifiable, and workload-balanced distributed algorithms working on a common input. An algorithm based on the framework is essentially a distributed encoding procedure for a Reed--Solomon code, which enables (a) robustness against byzantine failures with intrinsic error-correction and identification of failed nodes, and (b) independent randomized verification to check the entire computation for correctness, which takes essentially no more resources than each node individually contributes to the computation. The framework builds on recent Merlin--Arthur proofs of batch evaluation of Williams~[{\em Electron.\ Colloq.\ Comput.\ Complexity}, Report TR16-002, January 2016] with the observation that {\em Merlin's magic is not needed} for batch evaluation---mere Knights can prepare the proof, in parallel, and with intrinsic error-correction. The contribution of this paper is to show that in many cases the verifiable batch evaluation framework admits algorithms that match in total resource consumption the best known sequential algorithm for solving the problem. As our main result, we show that the $k$-cliques in an $n$-vertex graph can be counted {\em and} verified in per-node $O(n^{(\omega+\epsilon)k/6})$ time and space on $O(n^{(\omega+\epsilon)k/6})$ compute nodes, for any constant $\epsilon>0$ and positive integer $k$ divisible by $6$, where $2\leq\omega<2.3728639$ is the exponent of matrix multiplication. This matches in total running time the best known sequential algorithm, due to Ne{\v{s}}et{\v{r}}il and Poljak [{\em Comment.~Math.~Univ.~Carolin.}~26 (1985) 415--419], and considerably improves its space usage and parallelizability. Further results include novel algorithms for counting triangles in sparse graphs, computing the chromatic polynomial of a graph, and computing the Tutte polynomial of a graph.Comment: 42 p

Rank-width and Tree-width of H-minor-free Graphs

We prove that for any fixed r>=2, the tree-width of graphs not containing K_r as a topological minor (resp. as a subgraph) is bounded by a linear (resp. polynomial) function of their rank-width. We also present refinements of our bounds for other graph classes such as K_r-minor free graphs and graphs of bounded genus.Comment: 17 page

Contractors for flows

We answer a question raised by Lov\'asz and B. Szegedy [Contractors and connectors in graph algebras, J. Graph Theory 60:1 (2009)] asking for a contractor for the graph parameter counting the number of B-flows of a graph, where B is a subset of a finite Abelian group closed under inverses. We prove our main result using the duality between flows and tensions and finite Fourier analysis. We exhibit several examples of contractors for B-flows, which are of interest in relation to the family of B-flow conjectures formulated by Tutte, Fulkerson, Jaeger, and others.Comment: 22 pages, 1 figur

On infinite-finite duality pairs of directed graphs

The (A,D) duality pairs play crucial role in the theory of general relational structures and in the Constraint Satisfaction Problem. The case where both classes are finite is fully characterized. The case when both side are infinite seems to be very complex. It is also known that no finite-infinite duality pair is possible if we make the additional restriction that both classes are antichains. In this paper (which is the first one of a series) we start the detailed study of the infinite-finite case. Here we concentrate on directed graphs. We prove some elementary properties of the infinite-finite duality pairs, including lower and upper bounds on the size of D, and show that the elements of A must be equivalent to forests if A is an antichain. Then we construct instructive examples, where the elements of A are paths or trees. Note that the existence of infinite-finite antichain dualities was not previously known

A combinatorial proof of the extension property for partial isometries

We present a short and self-contained proof of the extension property for partial isometries of the class of all finite metric spaces.Comment: 7 pages, 1 figure. Minor revision. Accepted to Commentationes Mathematicae Universitatis Carolina