Related papers: On the Tutte and matching polynomials for complete…
We study the Tutte polynomial of two infinite families of finite graphs. These are the Schreier graphs associated with the action of two well-known self-similar groups acting on the binary rooted tree by automorphisms: the first Grigorchuk…
A graph $G$ is well-covered if all its maximal stable sets have the same size, denoted by alpha(G) (M. D. Plummer, 1970). If for any $k$ the $k$-th coefficient of a polynomial I(G;x) is equal to the number of stable sets of cardinality $k$…
In this chapter (Chapter V) we present several results which demonstrate a close connection and useful exchange of ideas between graph theory and knot theory. These disciplines were shown to be related from the time of Tait (if not Listing)…
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…
It is conjectured by Chen and Raspaud that for each integer $k \ge 2$, any graph $G$ with \[ \mathrm{mad}(G) < \frac{2k+1}{k} \quad\text{and}\quad \mathrm{odd\text{-}girth}(G) \ge 2k+1 \] admits a homomorphism into the Kneser graph…
A large body of research in graph theory concerns the induced subgraphs of graphs with large chromatic number, and especially which induced cycles must occur. In this paper, we unify and substantially extend results from a number of…
We associate a moduli problem to a colored trivalent graph; such graphs, when planar, appear in the state-sum description of the quantum sl(N) knot polynomial due to Murakami, Ohtsuki, and Yamada. We discuss how the resulting moduli space…
Inspired by the study of community structure in connection networks, we introduce the graph polynomial $Q(G;x,y)$, the bivariate generating function which counts the number of connected components in induced subgraphs. We give a recursive…
Jaeger, Vertigan, and Welsh [15] proved a dichotomy for the complexity of evaluating the Tutte polynomial at fixed points: The evaluation is #P-hard almost everywhere, and the remaining points admit polynomial-time algorithms. Dell,…
We prove that a large family of pairs of graphs satisfy a polynomial dependence in asymmetric graph removal lemmas. In particular, we give an unexpected answer to a question of Gishboliner, Shapira, and Wigderson by showing that for every…
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…
We study the computation of the Tutte polynomials of fan-like graphs and obtain expressions of their Tutte polynomials via generating functions. As applications, Tutte polynomials, in particular, the number of spanning trees, of two kinds…
Let $G=(V,E)$ be a simple graph with $V=\{1,2,\cdots,n\}$ and $\chi(G,x)$ be its chromatic polynomial. For an ordering $\pi=(v_1,v_2,\cdots,v_n)$ of elements of $V$, let $\delta_G(\pi)$ be the number of $i$'s, where $1\le i\le n-1$, with…
The Tutte polynomial is a fundamental invariant associated to a graph, matroid, vector arrangement, or hyperplane arrangement. This short survey focuses on some of the most important results on Tutte polynomials of hyperplane arrangements.…
We prove a formula for the ${\mathbb S}_n$-equivariant Euler characteristic of the moduli space of graphs $\mathcal{MG}_{g,n}$. Moreover, we prove that the rational ${\mathbb S}_n$-invariant cohomology of $\mathcal{MG}_{g,n}$ stabilizes for…
We prove that for every $t\in \mathbb{N}$ there is a constant $\gamma_t$ such that every graph with twin-width at most $t$ and clique number $\omega$ has chromatic number bounded by $2^{\gamma_t \log^{4t+3} \omega}$. In other words, we…
We present an algebraic characterization of perfect graphs, i.e., graphs for which the clique number and the chromatic number coincide for every induced subgraph. We show that a graph is perfect if and only if certain nonnegative…
We present exact calculations of Potts model partition functions and the equivalent Tutte polynomials for polygon chain graphs with open and cyclic boundary conditions. Special cases of the results that yield flow and reliability…
This paper studies Bernoulli cell complexes from the perspective of persistent homology, Tutte polynomials, and random-cluster models. Following the previous work [9], we first show the asymptotic order of the expected lifetime sum of the…
Thomassen, in 1983, conjectured that for a positive integer $k$, every $2$-connected non-bipartite graph of minimum degree at least $k + 1$ contains cycles of all lengths modulo $k$. In this paper, we settle this conjecture affirmatively.