Related papers: Tutte Short Exact Sequences of Graphs
Unlike Whitney's definition of the corank-nullity generating function $T(G;x+1,y+1)$, Tutte's definition of his now eponymous polynomial $T(G;x,y)$ requires a total order on the edges of which the polynomial is a posteriori independent.…
The Tutte polynomial is a classical invariant, important in combinatorics and statistical mechanics. An essential feature of the Tutte polynomial is the duality for planar graphs G, $T_G(X,Y)\; =\; {T}_{G^*}(Y,X)$ where $G^*$ denotes the…
Extending the work of Alon, Frieze abnd Welsh, we show that there are randomized polynomial time approximation schemes for computing the Tutte polynomial in subdense graphs with an minimal node degree of $\Omega\left ( \frac{n}{\sqrt{\log…
We give an analogue of the Tutte polynomial for hypermaps. This polynomial can be defined as either a sum over subhypermaps, or recursively through deletion-contraction reductions where the terminal forms consist of isolated vertices. Our…
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…
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…
A cycle $C$ in a graph $G$ is called a Tutte cycle if, after deleting $C$ from $G$, each component has at most three neighbors on $C$. Tutte cycles play an important role in the study of Hamiltonicity of planar graphs. Thomas and Yu and…
A generalization of Tutte polynomial involved in the evaluation of the moments of the integrated geometric Brownian in the Ito formalism is discussed. The new combinatorial invariant depends on the order in which the sequence of…
An edge subset $S$ of a connected graph $G$ is called an anti-Kekul\'{e} set if $G-S$ is connected and has no perfect matching. We can see that a connected graph $G$ has no anti-Kekul\'{e} set if and only if each spanning tree of $G$ has a…
A fourientation of a graph is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. Fixing a total order on the edges and a reference orientation of the graph, we…
A parking function of length n is a sequence (b_1, b_2,..., b_n) of nonnegative integers whose nondecreasing rearrangement (a_1, a_2,...,a_n) has the property that a_i < i for every i. A well-known result about parking functions is that the…
Originally in 1954 the Tutte polynomial was a bivariate polynomial associated to a graph in order to enumerate the colorings of this graph and of its dual graph at the same time. However the Tutte polynomial reveals more of the internal…
We offer a new structural basis for the theory of 3-connected graphs, providing a unique decomposition of every such graph into parts that are either quasi 4-connected, wheels, or thickened $K_{3,m}$'s. Our construction is explicit,…
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…
The classical relationship between the Tutte polynomial of graph theory and the Potts model of statistical mechanics has resulted in valuable interactions between the disciplines. Unfortunately, it does not include the external magnetic…
We present exact calculations of the $q$-state Potts model partition functions and the equivalent Tutte polynomials for chain graphs comprised of $m$ repeated hammock subgraphs $H_{e_1,...,e_r}$ connected with line graphs of length $e_g$…
We give a new characterization of the Tutte polynomial of graphs. Our characterization is formally close (but inequivalent) to the original definition given by Tutte as the generating function of spanning trees counted according to…
We derive exact relations between the Potts model partition function, or equivalently the Tutte polynomial, for a network (graph) $G$ and a network obtained from $G$ by (i) by replacing each edge (i.e., bond) of $G$ by two or more edges…
A graph $G$ is said to be $p$-periodic, if the automorphism group $Aut(G)$ contains an element of order $p$ which preserves no edges. In this paper, we investigate the behavior of graph polynomials (Negmai and Tutte) with respect to graph…
We combinatorially prove a new recurrence between the Tutte polynomials of graphs obtained by contraction of the complete graphs $K_{n}$%. This generalizes, to two variables, a relation previously obtained by the author between the…