We introduce a new two-player nonlocal game, called the (G, H)-isomorphism game, where classical players can win with certainty if and only if the graphs G and H are isomorphic. We then define the notions of quantum and non-signalling isomorphism, by considering perfect quantum and non-signalling strategies for the (G, H)-isomorphism game, respectively. First, we prove that non-signalling isomorphism coincides with the well-studied notion of fractional isomorphism, thus giving the latter an operational interpretation. In the quantum case, we consider both the tensor product and commuting frameworks for nonlocal games. Second, we show that, in the tensor product framework, quantum isomorphism is equivalent to the feasibility of two polynomial systems in non-commuting variables, obtained by relaxing the standard integer programming formulations for graph isomorphism to Hermitian variables. Finally, we provide a construction for reducing linear binary constraint system games to isomorphism games. This allows us to determine quantum isomorphic graphs that are not isomorphic. Furthermore, it allows us to show that our two notions of quantum isomorphism, from the tensor product and commuting frameworks, are in fact distinct relations, and that the latter is undecidable. Our proof techniques are related to the Feige, Goldwasser, Lovász, Safra, and Szegedy reduction from the inapproximability literature [

The star chromatic index χs′(G) of a graph G is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi‐colored. We obtain a near‐linear upper bound in terms of the maximum degree Δ=Δ(G). Our best lower bound on χnormals′ in terms of Δ is 2Δ(1+o(1)) valid for complete graphs. We also consider the special case of cubic graphs, for which we show that the star chromatic index lies between 4 and 7 and characterize the graphs attaining the lower bound. The proofs involve a variety of notions from other branches of mathematics and may therefore be of certain independent interest.

We introduce a two-player nonlocal game, called the (G, H)-isomorphism game, where classical players can win with certainty if and only if the graphs G and H are isomorphic. We then define the notions of quantum and non-signalling isomorphism, by considering perfect quantum and non-signalling strategies for the (G, H)-isomorphism game, respectively. In the quantum case, we consider both the tensor product and commuting frameworks for nonlocal games. We prove that non-signalling isomorphism coincides with the well-studied notion of fractional isomorphism, thus giving the latter an operational interpretation. Second, we show that, in the tensor product framework, quantum isomorphism is equivalent to the feasibility of two polynomial systems in non-commuting variables, obtained by relaxing the standard integer programming formulations for graph isomorphism to Hermitian variables. On the basis of this correspondence, we show that quantum isomorphic graphs are necessarily cospectral. Finally, we provide a construction for reducing linear binary constraint system games to isomorphism games. This allows us to produce quantum isomorphic graphs that are nevertheless not isomorphic. Furthermore, it allows us to show that our two notions of quantum isomorphism, from the tensor product and commuting frameworks, are in fact distinct relations, and that the latter is undecidable. Our construction is related to the FGLSS reduction from inapproximability literature, as well as the CFI construction.

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The Shortest Cycle Cover Conjecture of Alon and Tarsi asserts that the edges of every bridgeless graph with m edges can be covered by cycles of total length at most 7m/5 = 1.400m. We show that every cubic bridgeless graph has a cycle cover of total length at most 34m/21 ≈ 1.619m and every bridgeless graph with minimum degree three has a cycle cover of total length at most 44m/27 ≈ 1.630m.

123266 Discrete Comput Geom (2017) 58:265-292 In this work we focus on graph embeddings constructed using the eigenvectors of the least eigenvalue of the adjacency matrix of G, which we call least eigenvalue frameworks. We identify two necessary and sufficient conditions for such frameworks to be universally completable. Our conditions also allow us to give algorithms for determining whether a least eigenvalue framework is universally completable. Furthermore, our computations for Cayley graphs on Z n 2 (n ≤ 5) show that almost all of these graphs have universally completable least eigenvalue frameworks. In the second part of this work we study uniquely vector colorable (UVC) graphs, i.e., graphs for which the semidefinite program corresponding to the Lovász theta number (of the complementary graph) admits a unique optimal solution. We identify a sufficient condition for showing that a graph is UVC based on the universal completability of an associated framework. This allows us to prove that Kneser and q-Kneser graphs are UVC. Lastly, we show that least eigenvalue frameworks of 1-walk-regular graphs always provide optimal vector colorings and furthermore, we are able to characterize all optimal vector colorings of such graphs. In particular, we give a necessary and sufficient condition for a 1-walk-regular graph to be uniquely vector colorable.

Let A be a finite nonempty subset of an additive abelian group G, and let \Sigma(A) denote the set of all group elements representable as a sum of some subset of A. We prove that |\Sigma(A)| >= |H| + 1/64 |A H|^2 where H is the stabilizer of \Sigma(A). Our result implies that \Sigma(A) = Z/nZ for every set A of units of Z/nZ with |A| >= 8 \sqrt{n}. This consequence was first proved by Erd\H{o}s and Heilbronn for n prime, and by Vu (with a weaker constant) for general n.Comment: 12 page

Stanisław Ulam asked whether there exists a universal countable planar graph (that is, a countable planar graph that contains every countable planar graph as a subgraph). János Pach (1981) answered this question in the negative. We strengthen this result by showing that every countable graph that contains all countable planar graphs must contain (i) an infinite complete graph as a minor, and (ii) a subdivision of the complete graph K t with multiplicity t, for every finite t.On the other hand, we construct a countable graph that contains all countable planar graphs and has several key properties such as linear colouring numbers, linear expansion, and every finite n-vertex subgraph has a balanced separator of size O( √ n). The graph is T 6 P ∞ , where T k is the universal treewidth-k countable graph (which we define explicitly), P ∞ is the 1-way infinite path, and denotes the strong product. More generally, for every positive integer t we construct a countable graph that contains every countable K t -minor-free graph and has the above key properties.Our final contribution is a construction of a countable graph that contains every countable K t -minor-free graph as an induced subgraph, has linear colouring numbers and linear expansion, and contains no subdivision of the countably infinite complete graph (implying (ii) above is best possible).