5,481 research outputs found

    Extremal Infinite Graph Theory

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    We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure

    Algebraic matroids with graph symmetry

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    This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a) algebraic matroids, we expose cryptomorphisms making them accessible to techniques from commutative algebra. This allows us to introduce for each circuit in an algebraic matroid an invariant called circuit polynomial, generalizing the minimal poly- nomial in classical Galois theory, and studying the matroid structure with multivariate methods. For (b) matroids with symmetries we introduce combinatorial invariants capturing structural properties of the rank function and its limit behavior, and obtain proofs which are purely combinatorial and do not assume algebraicity of the matroid; these imply and generalize known results in some specific cases where the matroid is also algebraic. These results are motivated by, and readily applicable to framework rigidity, low-rank matrix completion and determinantal varieties, which lie in the intersection of (a) and (b) where additional results can be derived. We study the corresponding matroids and their associated invariants, and for selected cases, we characterize the matroidal structure and the circuit polynomials completely

    On First-Order Definable Colorings

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    We address the problem of characterizing HH-coloring problems that are first-order definable on a fixed class of relational structures. In this context, we give several characterizations of a homomorphism dualities arising in a class of structure

    Branch-depth: Generalizing tree-depth of graphs

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    We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph G=(V,E)G = (V,E) and a subset AA of EE we let λG(A)\lambda_G (A) be the number of vertices incident with an edge in AA and an edge in EAE \setminus A. For a subset XX of VV, let ρG(X)\rho_G(X) be the rank of the adjacency matrix between XX and VXV \setminus X over the binary field. We prove that a class of graphs has bounded tree-depth if and only if the corresponding class of functions λG\lambda_G has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions ρG\rho_G has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by the restriction.Comment: 34 pages, 2 figure

    Branch-depth: Generalizing tree-depth of graphs

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    We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph G=(V,E)G = (V,E) and a subset AA of EE we let λG(A)\lambda_G (A) be the number of vertices incident with an edge in AA and an edge in EAE \setminus A. For a subset XX of VV, let ρG(X)\rho_G(X) be the rank of the adjacency matrix between XX and VXV \setminus X over the binary field. We prove that a class of graphs has bounded tree-depth if and only if the corresponding class of functions λG\lambda_G has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions ρG\rho_G has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by the restriction.Comment: 36 pages, 2 figures. Final versio
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