349 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
Upper oriented chromatic number of undirected graphs and oriented colorings of product graphs
The oriented chromatic number of an oriented graph is the minimum
order of an oriented graph \vev H such that admits a homomorphism to
\vev H. The oriented chromatic number of an undirected graph is then the
greatest oriented chromatic number of its orientations. In this paper, we
introduce the new notion of the upper oriented chromatic number of an
undirected graph , defined as the minimum order of an oriented graph \vev
U such that every orientation of admits a homomorphism to . We give some properties of this parameter, derive some general upper bounds
on the ordinary and upper oriented chromatic numbers of Cartesian, strong,
direct and lexicographic products of graphs, and consider the particular case
of products of paths.Comment: 14 page
Set maps, umbral calculus, and the chromatic polynomial
Some important properties of the chromatic polynomial also hold for any
polynomial set map satisfying p_S(x+y)=\sum_{T\uplus U=S}p_T(x)p_U(y). Using
umbral calculus, we give a formula for the expansion of such a set map in terms
of any polynomial sequence of binomial type. This leads to some new expansions
of the chromatic polynomial. We also describe a set map generalization of Abel
polynomials.Comment: 20 page
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