Related papers: Acyclic orientations and poly-Bernoulli numbers
In this work we study the acyclic orientations of graphs. We obtain an encoding of the acyclic orientations of the complete $p$-partite graph with size of its parts $n_1,n_2,\ldots,n_p$ via a vector with $p$ symbols and length…
Building on previous work by Cameron et al. in [3], we give a recurrence for computing the number of acyclic orientations of complete $k$-partite graphs, which can be implemented to obtain a dynamic programming algorithm running in time…
Gessel and Sagan investigated the Tutte polynomial, $T(x,y)$ using depth first search, and applied their techniques to show that the number of acyclic partial orientations of a graph is $2^gT(3,1/2)$. We provide a short deletion-contraction…
In this paper, we study oriented bipartite graphs. In particular, we introduce "bitransitive" graphs. Several characterizations of bitransitive bitournaments are obtained. We show that bitransitive bitounaments are equivalent to acyclic…
We define the acyclic orientation polynomial of a graph to be the generating function for the sinks of its acyclic orientations. Stanley proved that the number of acyclic orientations is equal to the chromatic polynomial evaluated at $-1$…
The ${\mathbb B}_n^{(k)}$ poly-Bernoulli numbers --- a natural generalization of classical Bernoulli numbers ($B_n={\mathbb B}_n^{(1)}$) --- were introduced by Kaneko in 1997. When the parameter $k$ is negative then ${\mathbb B}_n^{(k)}$ is…
The permanental polynomial of a graph $G$ is $\pi(G,x)\triangleq\mathrm{per}(xI-A(G))$. From the result that a bipartite graph $G$ admits an orientation $G^e$ such that every cycle is oddly oriented if and only if it contains no even…
We study the problem of finding an acyclic orientation of an undirected graph with constrained in-degree parities specified by a subset T of vertices. An orientation is called T -odd if a vertex v has odd in-degree if and only if v P T .…
Randomly sampling an acyclic orientation on the complete bipartite graph $K_{n,k}$ with parts of size $n$ and $k$, we investigate the length of the longest path. We provide a probability generating function for the distribution of the…
We present a new correspondence between acyclic orientations and coloring of a signed graph (symmetric graph). Goodall et al. introduced a bivariate chromatic polynomial $\chi_G(k,l)$ that counts the number of signed colorings using colors…
Recall that an excedance of a permutation $\pi$ is any position $i$ such that $\pi_i > i$. Inspired by the work of Hopkins, McConville and Propp (Elec. J. Comb., 2017) on sorting using toppling, we say that a permutation is toppleable if it…
This paper solves a problem that was stated by M. A. Harrison in 1973~\cite{harrison1973number}. This problem, that has remained open since then is concerned with counting equivalence classes of $n\times r$ binary matrices under row and…
Let $I$ and $O$ denote two sets of vertices, where $I\cap O =\emptyset$, $|I| = n$, $|O| = r$, and $B_u(n,r)$ denote the set of unlabeled graphs whose edges connect vertices in $I$ and $O$. Recently, it was established…
Greene and Zaslavsky proved that the number of acyclic orientations of a graph with a unique sink is, up to sign, the linear coefficient of the chromatic polynomial. We give three new proofs of this result using pure induction,…
We consider, for complete bipartite graphs, the convex hulls of characteristic vectors of all matchings, extended by a binary entry indicating whether the matching contains two specific edges. These polytopes are associated to the quadratic…
An acyclic edge-coloring of a graph is a proper edge-coloring without bichromatic ($2$-colored) cycles. The acyclic chromatic index of a graph $G$, denoted by $a'(G)$, is the least integer $k$ such that $G$ admits an acyclic edge-coloring…
We show that the number of $k$-matching in a given undirected graph $G$ is equal to the number of perfect matching of the corresponding graph $G_k$ on an even number of vertices divided by a suitable factor. If $G$ is bipartite then one can…
The chromatic polynomial $P(G,x)$ of a graph $G$ of order $n$ can be expressed as $\sum\limits_{i=1}^n(-1)^{n-i}a_{i}x^i$, where $a_i$ is interpreted as the number of broken-cycle free spanning subgraphs of $G$ with exactly $i$ components.…
A polynomial-time exact algorithm for counting the number of directed acyclic graphs in a Markov equivalence class was recently given by Wien\"obst, Bannach, and Li\'skiewicz (AAAI 2021). In this paper, we consider the more general problem…
The aim of this paper is to show that any finite undirected bipartite graph can be considered as a polynomial $p \in \mathbb{N}[x]$, and any directed finite bipartite graph can be considered as a polynomial $p\in\mathbb{N}[x,y]$, and vise…