5,481 research outputs found
Extremal Infinite Graph Theory
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
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
We address the problem of characterizing -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
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 and a subset of we let
be the number of vertices incident with an edge in and an
edge in . For a subset of , let be the rank
of the adjacency matrix between and over the binary field.
We prove that a class of graphs has bounded tree-depth if and only if the
corresponding class of functions has bounded branch-depth and
similarly a class of graphs has bounded shrub-depth if and only if the
corresponding class of functions 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
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 and a subset of we let
be the number of vertices incident with an edge in and an
edge in . For a subset of , let be the rank
of the adjacency matrix between and over the binary field.
We prove that a class of graphs has bounded tree-depth if and only if the
corresponding class of functions has bounded branch-depth and
similarly a class of graphs has bounded shrub-depth if and only if the
corresponding class of functions 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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