Related papers: Permutation Tutte polynomial

We prove a conjecture of Shteiner and Shteyner stating that for a bipartite graph $G=(V,E)$, the number of forests in $G$ equals the number of degree sequences arising from its spanning subgraphs. In the process, we provide several…

We continue studying Thomassen's conjecture (every 4-connected line graph has a Hamilton cycle) in the direction of a recently shown equivalence with Jackson's conjecture (every 2-connected claw-free graph has a Tutte cycle), and we extend…

Many graph polynomials, such as the Tutte polynomial, the interlace polynomial and the matching polynomial, have both a recursive definition and a defining subset expansion formula. In this paper we present a general, logic-based framework…

This paper introduces a conceptual framework, in the context of quantum topology and the algebras underlying it, for analyzing relations obeyed by the chromatic polynomial \chi(Q) of planar graphs. Using it we give new proofs and…

To a singular knot K with n double points, one can associate a chord diagram with n chords. A chord diagram can also be understood as a 4-regular graph endowed with an oriented Euler circuit. L. Traldi introduced a polynomial invariant for…

Tutte showed that a graph $G$ is planar if and only if the conflict graph associated to every cycle of $G$ is bipartite. We define a (not necessarily unique) signed conflict graph associated to a maximally planar subgraph of a nonplanar…

We introduce the notions of arithmetic colorings and arithmetic flows over a graph with labelled edges, which generalize the notions of colorings and flows over a graph. We show that the corresponding arithmetic chromatic polynomial and…

It is well-known that every planar graph has a Tutte path, i.e., a path $P$ such that any component of $G-P$ has at most three attachment points on $P$. However, it was only recently shown that such Tutte paths can be found in polynomial…

Matroid theory is often thought of as a generalization of graph theory. In this paper we propose an analogous correspondence between embedded graphs and delta-matroids. We show that delta-matroids arise as the natural extension of graphic…

The interior polynomial is an invariant of (signed) bipartite graphs, and the interior polynomial of a plane bipartite graph is equal to a part of the HOMFLY polynomial of a naturally associated link. The HOMFLY polynomial $P_L(v,z)$ is a…

Stanley's symmetrized chromatic polynomial is a generalization of the ordinary chromatic polynomial to a graph invariant with values in a ring of polynomials in infnitely many variables. The ordinary chromatic polynomial is a specialization…

The {\em bipartite-hole-number} of a graph $G$, denoted as $\widetilde\alpha(G)$, is the minimum number $k$ such that there exist integers $a$ and $b$ with $a + b = k+1$ such that for any two disjoint sets $A, B \subseteq V(G)$, there is an…

In this paper, we compare the $\mathrm{v}$-numbers and the degree of the $h$-polynomials associated with edge ideals of connected graphs. We prove that the $\mathrm{v}$-number can be arbitrarily larger or smaller than the degree of the…

We introduce a new bivariate polynomial ${\displaystyle J(G; x,y):=\sum\limits_{W \in V(G)} x^{|W|}y^{|N(W)|}}$ which contains the standard domination polynomial of the graph $G$ in two different ways. We build methods for efficient…

The periodic discrete Toda equation defined over finite fields has been studied. We obtained the finite graph structures constructed by the network of states where edges denote possible time evolutions. We simplify the graphs by introducing…

Let $G$ be a finite simple graph and let $I_G$ denote its associated toric ideal in the polynomial ring $R$. For each integer $n\geq 2$, we completely determine all the possible values for the tuple $({\rm reg}(R/I_G), {\rm…

Following Britz, Johnsen, Mayhew and Shiromoto, we consider demi\-ma\-troids as a(nother) natural generalization of matroids. As they have shown, demi\-ma\-troids are the appropriate combinatorial objects for studying Wei's duality. Our…

This paper deals with the partition function of the Ising model from statistical mechanics, which is used to study phase transitions in physical systems. A special case of interest is that of the Ising model with constant energies and…

The Poincar\'e polynomial of the complement of an arrangements in a non compact group is a specialization of the $G$-Tutte polynomial associated with the arrangement. In this article we show two unimodular elliptic arrangements (built up…

Let $e$ be a positive integer, $p$ be an odd prime, $q=p^{e}$, and $\Bbb F_q$ be the finite field of $q$ elements. Let $f,g \in \Bbb F_q [X,Y]$. The graph $G=G_q(f,g)$ is a bipartite graph with vertex partitions $P=\Bbb F_q^3$ and $L=\Bbb…