2,198 research outputs found
Connection Matrices and the Definability of Graph Parameters
In this paper we extend and prove in detail the Finite Rank Theorem for
connection matrices of graph parameters definable in Monadic Second Order Logic
with counting (CMSOL) from B. Godlin, T. Kotek and J.A. Makowsky (2008) and
J.A. Makowsky (2009). We demonstrate its vast applicability in simplifying
known and new non-definability results of graph properties and finding new
non-definability results for graph parameters. We also prove a Feferman-Vaught
Theorem for the logic CFOL, First Order Logic with the modular counting
quantifiers
Colorings, determinants and Alexander polynomials for spatial graphs
A {\em balanced} spatial graph has an integer weight on each edge, so that
the directed sum of the weights at each vertex is zero. We describe the
Alexander module and polynomial for balanced spatial graphs (originally due to
Kinoshita \cite{ki}), and examine their behavior under some common operations
on the graph. We use the Alexander module to define the determinant and
-colorings of a balanced spatial graph, and provide examples. We show that
the determinant of a spatial graph determines for which the graph is
-colorable, and that a -coloring of a graph corresponds to a
representation of the fundamental group of its complement into a metacyclic
group . We finish by proving some properties of the Alexander
polynomial.Comment: 14 pages, 7 figures; version 3 reorganizes the paper, shortens some
of the proofs, and improves the results related to representations in
metacyclic groups. This is the final version, accepted by Journal of Knot
Theory and its Ramification
A bivariate chromatic polynomial for signed graphs
We study Dohmen--P\"onitz--Tittmann's bivariate chromatic polynomial
which counts all -colorings of a graph such
that adjacent vertices get different colors if they are . Our first
contribution is an extension of to signed graphs, for which we
obtain an inclusion--exclusion formula and several special evaluations giving
rise, e.g., to polynomials that encode balanced subgraphs. Our second goal is
to derive combinatorial reciprocity theorems for and its
signed-graph analogues, reminiscent of Stanley's reciprocity theorem linking
chromatic polynomials to acyclic orientations.Comment: 8 pages, 4 figure
Algebraic Characterization of Uniquely Vertex Colorable Graphs
The study of graph vertex colorability from an algebraic perspective has
introduced novel techniques and algorithms into the field. For instance, it is
known that -colorability of a graph is equivalent to the condition for a certain ideal I_{G,k} \subseteq \k[x_1, ..., x_n]. In this
paper, we extend this result by proving a general decomposition theorem for
. This theorem allows us to give an algebraic characterization of
uniquely -colorable graphs. Our results also give algorithms for testing
unique colorability. As an application, we verify a counterexample to a
conjecture of Xu concerning uniquely 3-colorable graphs without triangles.Comment: 15 pages, 2 figures, print version, to appear J. Comb. Th. Ser.
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