Totally frustrated states in the chromatic theory of gain graphs

The Chromatic Polynomial of a Digraph

An acyclic coloring of a digraph as defined by Neumann-Lara is a vertex-coloring such that no monochromatic directed cycles occur. Counting the number of such colorings with $k$ colors can be done by counting so-called Neumann-Lara-coflows (NL-coflows), which build a polynomial in $k$. We will present a representation of this polynomial using totally cyclic subdigraphs, which form a graded poset $Q$. Furthermore we will decompose our NL-coflow polynomial, which becomes the chromatic polynomial of a digraph by multiplication with the number of colors to the number of components, examining the s…

Kneser 图中的全奇分拆

Totally odd subdivisions in Kneser graphs

本文加强了 Kneser 图中关于 Odd Hadwiger Conjecture 的已有结果，证明了存在包含任意大完全全奇分拆的 $t$-色 Kneser 图，且所有 Kneser 图均含 $K_{χ(G)}$ 的全奇分拆，并扩展至全奇浸入研究。

Properties of chromatic polynomials of hypergraphs not held for chromatic polynomials of graphs

In this paper, we present some properties on chromatic polynomials of hypergraphs which do not hold for chromatic polynomials of graphs. We first show that chromatic polynomials of hypergraphs have all integers as their zeros and contain dense real zeros in the set of real numbers. We then prove that for any multigraph $G=(V,E)$, the number of totally cyclic orientations of $G$ is equal to the value of $|P(H,-1)|$, where $P(H,λ)$ is the chromatic polynomial of a hypergraph $H$ which is constructed from $G$. Finally we show that the multiplicity of root "$0$" of $P(H,λ)$ may be at least $2$ for…

Borel chromatic numbers of locally countable $F_σ$ graphs and forcing with superperfect trees

In this work we study the uncountable Borel chromatic numbers, defined by Geschke (2011) as cardinal characteristics of the continuum, of low complexity graphs. We show that a strong form of locally countable graphs with compact totally disconnected set of vertices have Borel chromatic number bounded by the continuum of the ground model in the model obtained by adding $\aleph_2$ Laver reals. From this, we answer a question from Geschke and the second author (2022), and another question from Fisher, Friedman and Khomskii (2014) concerning regularity properties of subsets of the real line.

最大度为8的平面图是全9-可着色的一个充分条件

A sufficient condition for planar graphs with maximum degree eight to be totally 9-colorable

针对最大度为8的平面图的全着色问题，证明了不含4-扇子图的此类图全色数为9，给出了平面图全9-可着色的一个充分条件。

Enumerating Colorings, Tensions and Flows in Cell Complexes

We study quasipolynomials enumerating proper colorings, nowhere-zero tensions, and nowhere-zero flows in an arbitrary CW-complex $X$, generalizing the chromatic, tension and flow polynomials of a graph. Our colorings, tensions and flows may be either modular (with values in $\mathbb{Z}/k\mathbb{Z}$ for some $k$) or integral (with values in $\{-k+1,\dots,k-1\}$). We obtain deletion-contraction recurrences and closed formulas for the chromatic, tension and flow quasipolynomials, assuming certain unimodularity conditions. We use geometric methods, specifically Ehrhart theory and inside-out polyto…