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The notion of thin sums matroids was invented to extend the notion of representability to non-finitary matroids. A matroid is tame if every circuit-cocircuit intersection is finite. |
Consider a card guessing game with complete feedback in which a deck of $n$ cards ordered $1,\dots, n$ is riffle-shuffled once. With the goal to maximize the number of correct guesses, a player guesses cards from the top of the deck one at a time under the optimal strategy until no cards remain. |
We give a generalization and a short mechanized proof of determinant conjectured by G. Kuperberg and J. Propp. <br>Further generalizations and applications of the method to some q-analogues may be found in <a href="http://www.math.temple.edu/~tewodros" rel="external noopener nofollow" class="link-external link-http">t... |
For a weighted digraph without loops $V$, the arc weights of which can be obtained from an undirected graph with loops ${\sf P}$ according to the rule $v_{ij}=p_{ij}-p_{ii}$, the properties are studied. An effective algorithm for constructing directed trees of minimum weight and an algorithm for constructing spanning ... |
We give a complete solution to the extremal topological combinatorial problem of finding the minimum number of tiles needed to construct a polyomino with $h$ holes. We denote this number by $g(h)$ and say that a polyomino is crystallized if it has $h$ holes and $g(h)$ tiles. |
We give an example of two $n\times n$ chess positions, $A$ and $B$, such that (1) there is a sequence $\sigma$ of legal chess moves leading from $A$ to $B$; (2) the length of $\sigma$ cannot be less than $\exp \Theta(n)$. |
We consider an integral series f(X,t) which depends on the choice of a set X of labelled planar rooted trees. We prove that its inverse for composition is of the form f(Z,t) for another set Z of trees, deduced from X. The proof is self-contained, though inspired by the Koszul duality theory of quadratic operads. |
To any Schur polynomial $s_{\lambda}$ one can associated its derived polynomials $s_{\lambda}{(i)}$ $i=0,\ldots,|\lambda|$ by the rule $$s_{\lambda}(x_1+t,\ldots,x_n+t) = \sum_i s_{\lambda}^{(i)}(x_1,\ldots,x_n) t^i. $$ We conjecture that $$(s_{\lambda}^{(i)})^2 - s_{\lambda}^{(i-1)} s_{\lambda}^{(i+1)}$$ is always Sch... |
A finite abstract simplicial complex G defines the Barycentric refinement graph phi(G) = (G,{ (a,b), a subset b or b subset a }) and the connection graph psi(G) = (G,{ (a,b), a intersected with b not empty }). We note here that both functors phi and psi from complexes to graphs are invertible on the image (Theorem 1) ... |
A transitive decomposition of a graph is a partition of the edge set together with a group of automorphisms which transitively permutes the parts. In this paper we determine all transitive decompositions of the Johnson graphs such that the group preserving the partition is arc-transitive and acts primitively on the pa... |
We prove various results on the size and structure of subsets of vector spaces over finite fields which, in some sense, have too many mutually orthogonal pairs of vectors. In particular, we obtain sharp finite field variants of a theorem of Rosenfeld and an almost version of a theorem of Berlekamp. |
MacPherson conjectured that the Grassmannian $\mathrm{Gr}(2, \mathbb{R}^n)$ has the same homeomorphism type as the combinatorial Grassmannian $\|\mbox{MacP}(2,n)\|$, while Babson proved that the spaces $\mathrm{Gr}(2,\mathbb{R}^n)$ and $\mathrm{Gr}(1,2,\mathbb{R}^n)$ are homotopy equivalent to their combinatorial analo... |
In this paper, we consider the family of pattern-replacement equivalence relations referred to as the "indices and values adjacent" case. Each such equivalence is determined by a partition $P$ of a subset of $S_c$ for some $c$. |
We prove that the generalized moment-cumulant relations introduced in [<a href="https://arxiv.org/abs/1711.00219" data-arxiv-id="1711.00219" class="link-https">arXiv:1711.00219</a>] are given by the action of the Eulerian idempotents on the Solomon-Tits algebras, whose direct sum builds up the Hopf algebra of Word Quas... |
A subfamily $\{F_1,F_2,\dots,F_{|P|}\}\subseteq \mathcal F$ is a copy of the poset $P$ if there exists a bijection $i:P\rightarrow \{F_1,F_2,\dots,F_{|P|}\}$ such that $p\le_P q$ implies $i (p)\subseteq i |
We prove that every 3-connected planar graph on $n$ vertices contains an induced path on $\Omega(\log n)$ vertices, which is best possible and improves the best known lower bound by a multiplicative factor of $\log \log n$. We deduce that any planar graph (or more generally, any graph embeddable on a fixed surface) wi... |
Given a finite abelian group $G$ and a subset $J\subset G$ with $0\in J$, let $D_{G}(J,N)$ be the maximum size of $A\subset G^{N}$ such that the difference set $A-A$ and $J^{N}$ have no non-trivial intersection. Recently, this extremal problem has been widely studied for different groups $G$ and subsets $J$. |
In 2001, J. -M. |
The de Bruijn torus (or grid) problem looks to find an $n$-by-$m$ binary matrix in which every possible $j$-by-$k$ submatrix appears exactly once. The existence and construction of these binary matrices was determined in the 70's, with generalizations to $d$-ary matrices in the 80's and 90's. |
In this paper we give a new characterization of the dual polar graphs, extending the work of Brouwer and Wilbrink on regular near polygons. Also as a consequence of our characterization we confirm a conjecture of the authors on non-bipartite distance-regular graphs with smallest eigenvalue at most $-k/2$, where $k$ is... |
The tilings of the 2-dimensional sphere by congruent triangles have been extensively studied, and the edge-to-edge tilings have been completely classified. However, not much is known about the tilings by other congruent polygons. |
Let $st=\{st_1,\ldots,st_k\}$ be a set of $k$ statistics on permutations with $k\geq 1$. We say that two given subset of permutations $T$ and $T'$ are $st$-Wilf-equivalent if the joint distributions of all statistics in $st$ over the sets of $T$-avoiding permutations $S_n(T)$ and $T'$-avoiding permutations $S_... |
We conjecture that finite graphs with positive Cheeger constant admit a spanning subgraph with positive Cheeger constant and girth proportional to the diameter. We prove this conjecture for regular expander graphs with large expansion. |
In a recent paper, Kahn gave the strongest possible, affirmative, answer to Shamir's problem, which had been open since the late 1970s: Let $r \ge 3 $ and let $n$ be divisible by $r$. Then, in the random $r$-uniform hypergraph process on $n$ vertices, as soon as the last isolated vertex disappears, a perfect match... |
In the present paper we show that the spectrum of an arbitrary starlike graph can be completely determined via separating functions $\rho_t$ (see \cite{NazRoi,RedRoi,Red3}). This fact helps to get in an easy way several results for the spectra of starlike graphs. |
The Erdős-Simonovits stability theorem is one of the most widely used theorems in extremal graph theory. We obtain an Erdős-Simonovits type stability theorem in multi-partite graphs. |
In 2017, Aharoni proposed the following generalization of the Caccetta-Häggkvist conjecture: if $G$ is a simple $n$-vertex edge-colored graph with $n$ color classes of size at least $r$, then $G$ contains a rainbow cycle of length at most $\lceil n/r \rceil$. <br>In this paper, we prove that, for fixed $r$, Aharoni... |
Let $p_{k,3}(n)$ enumerate the number of 2-color partition triples of $n$ where one of the colors appears only in parts that are multiples of $k$. In this paper, we prove several infinite families of congruences modulo powers of 3 for $p_{k,3}(n)$ with $k=1, 3$, and $9$. |
Borrowing inspiration from Marcone and Montálban's one-one correspondence between the class of signed trees and the equimorphism classes of indecomposable scattered linear orders, we find a subclass of signed trees which has an analogous correspondence with equimorphism classes of indecomposable finite rank discret... |
We introduce the immersion poset $(\mathcal{P}(n), \leqslant_I)$ on partitions, defined by $\lambda \leqslant_I \mu$ if and only if $s_\mu(x_1, \ldots, x_N) - s_\lambda(x_1, \ldots, x_N)$ is monomial-positive. Relations in the immersion poset determine when irreducible polynomial representations of $GL_N(\mathbb{C})$ ... |
In this paper, we give a representation of the average of complete joint weight enumerators of two linear codes of length $n$ over $\mathbb{F}_{q}$ and $\mathbb{Z}_{k}$ in terms of the compositions of $n$ and their distributions in the codes. We also obtain a generalization of the representation for the average of $g$... |
A $k\ell$-subset partition, or $(k,\ell)$-subpartition, is a $k\ell$-subset of an $n$-set that is partitioned into $\ell$ distinct classes, each of size $k$. Two $(k,\ell)$-subpartitions are said to $t$-intersect if they have at least $t$ classes in common. |
We obtain results on the limiting distribution of the six-length of a random functional graph, also called a functional digraph or random mapping, with given in-degree sequence. The six-length of a vertex $v\in V$ is defined from the associated mapping, $f:V\to V$, to be the maximum $i\in V$ such that the elements $v,... |
Let $G$ be a finite simple graph on $n$ vertices, that contains no isolated vertices, and let $I(G) \subseteq S = K[x_1, \dots, x_n]$ be its edge ideal. In this paper, we study the pair of integers that measure the projective dimension and the regularity of $S/I(G)$. |
In this paper we study the class of graphs $G_{m,n}$ that have the same degree sequence as two disjoint cliques $K_m$ and $K_n$, as well as the class $\overline G_{m,n}$ of the complements of such graphs. We establish various properties of $G_{m,n}$ and $\overline G_{m,n}$ related to recognition, connectivity, diamete... |
we define the sunlet graph is the graph obtained by taking one copy of the cycle and joined every vertex of the cycle to an exactly one pendant vertex such that the degree of each vertex in the cycle is three. In this paper, we establish necessary and sufficient conditions for the existence of decomposition of the Car... |
A generalization of list-coloring, now known as DP-coloring, was recently introduced by Dvořák and Postle. Essentially, DP-coloring assigns an arbitrary matching between lists of colors at adjacent vertices, as opposed to only matching identical colors as is done for list-coloring. |
The classical theorem due to Győri and Lovász states that any $k$-connected graph $G$ admits a partition into $k$ connected subgraphs, where each subgraph has a prescribed size and contains a prescribed vertex, as long as the total size of target subgraphs is equal to the size of $G$. However, this result is notorious... |
Tittmann, Averbouch and Makowsky [P. Tittmann, I. Averbouch, J.A. Makowsky, The enumeration of vertex induced subgraphs with respect to the number of components, European Journal of Combinatorics, 32 (2011) 954-974], introduced the subgraph component polynomial $Q(G;x,y)$ which counts the number of connected components... |
We study the abelian sandpile growth model, where n particles are added at the origin on a stable background configuration in Z^d. Any site with at least 2d particles then topples by sending one particle to each neighbor. |
Let $X$ be a list of vectors that is totally unimodular. In a previous article the author proved that every real-valued function on the set of interior lattice points of the zonotope defined by $X$ can be extended to a function on the whole zonotope of the form $p(D)B_X$ in a unique way, where $p(D)$ is a differential... |
Let $G$ be a fixed graph. Two paths of length $n-1$ on $n$ vertices (Hamiltonian paths) are $G$-different if there is a subgraph isomorphic to $G$ in their union. |
A common model of robustness of a graph against random failures has all vertices operational, but the edges independently operational with probability $p$. One can ask for the probability that all vertices can communicate ({\em all-terminal reliability}) or that two specific vertices (or {\em terminals}) can communica... |
Let $\pi:{\mathbb R}^n \to {\mathbb R}^d$ be any linear projection, let $A$ be the image of the standard basis. Motivated by Postnikov's study of postitive Grassmannians via plabic graphs and Galashin's connection of plabic graphs to slices of zonotopal tilings of 3-dimensional cyclic zonotopes, we study the p... |
We give a proof of a conjecture of A. Lacasse in his doctoral thesis which has applications in machine learning algorithms. The proof relies on some interesting binomial sums identities introduced by Abel (1839), and on their generalization to the multinomial case by Hurwitz (1902). |
In his seminal paper from 1952 Dirac showed that the complete graph on $n\geq 3$ vertices remains Hamiltonian even if we allow an adversary to remove $\lfloor n/2\rfloor$ edges touching each vertex. In 1960 Ghouila-Houri obtained an analogue statement for digraphs by showing that every directed graph on $n\geq 3$ vert... |
Words are sequences of letters over a finite alphabet. We study two intimately related topics for this object: quasi-randomness and limit theory. |
We relate the linear asymptotic representation theory of the symmetric groups to its spin counterpart. In particular, we give explicit formulas which express the normalized irreducible spin characters evaluated on a strict partition $\xi$ with analogous normalized linear characters evaluated on the double partition $D... |
We study the balance of $G$-gain graphs, where $G$ is an arbitrary group, by investigating their adjacency matrices and their spectra. As a first step, we characterize switching equivalence and balance of gain graphs in terms of their adjacency matrices in $M_n(\mathbb C G)$. |
Kitaev, Potapov, and Vajnovszki [On shortening u-cycles and u-words for permutations, Discrete Appl. Math, 2019] described how to shorten universal words for permutations, to length $n! +n-1-i(n-1)$ for any $i \in [(n-2)!]$, by introducing incomparable elements. |
We prove that if $P$ is a set of $n$ points in $\mathbb{C}^2$, then either the points in $P$ determine $\Omega(n^{1-\epsilon})$ complex distances, or $P$ is contained in a line with slope $\pm i$. If the latter occurs then each pair of points in $P$ have complex distance 0. |
A tree $T$ in an edge-colored graph $H$ is called a \emph{monochromatic tree} if all the edges of $T$ have the same color. For $S\subseteq V(H)$, a \emph{monochromatic $S$-tree} in $H$ is a monochromatic tree of $H$ containing the vertices of $S$. |
We give a simple proof of the property discovered by Xavier Grandsart: let $W$ be a circular binary word; then the differences in the number of occurences $|W|_{0011}-|W|_{1100}$, $ |W|_{1101}-|W|_{1011}$ , $|W|_{1010}-|W|_{0101}$ and $|W|_{0100}-|W|_{0010}$ are equal. |
Let $R$ be a commutative ring with unity. The weakly zero-divisor graph $W\Gamma(R)$ of the ring $R$ is the simple undirected graph whose vertices are nonzero zero-divisors of $R$ and two vertices $x$, $y$ are adjacent if and only if there exists $r\in {\rm ann} |
Let $A$ be a real matrix. The term rank of $A$ is the smallest number $t$ of lines (that is, rows or columns) needed to cover all the nonzero entries of $A$. |
An edge-colored graph $G$ is rainbow connected, if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. |
For a graph $G$ and integer $k\geq1$, we define the token graph $F_k(G)$ to be the graph with vertex set all $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference is a pair of adjacent vertices in $G$. Thus vertices of $F_k(G)$ correspond to configurations of $k$ indist... |
Erdős and Hanani proved that for every fixed integer $k \ge 2$, the complete graph $K_n$ can be almost completely packed with copies of $K_k$; that is, $K_n$ contains pairwise edge-disjoint copies of $K_k$ that cover all but an $o_n(1)$ fraction of its edges. Equivalently, elements of the set $\C(k)$ of all red-blue e... |
The study of graph drawings on 2-surfaces is an active area of mathematical research. Our main results are criteria for integer and modulo 2 embeddability of graphs to surfaces. |
By Menger's theorem the maximum number of arc-disjoint paths from a vertex s to a vertex t in a directed graph equals the minumum number of arcs needed to disconnect s and t, i.e., the minimum size of an s-t-cut. The max-flow problem in a network with unit capacities is equivalent to the arc-disjoint paths problem... |
We prove a conjecture of Stembridge concerning stability of Kronecker coefficients that vastly generalizes Murnaghan's theorem. The main idea is to identify the sequences of Kronecker coefficients in question with Hilbert functions of modules over finitely generated algebras. |
Let $G=(V(G),E(G))$ be a simple graph, where $V(G)$ and $E(G)$ are the vertex set and the edge set of $G$, respectively. The number of components of $G$ is denoted by $c(G)$. |
We investigate the relationship between two constructions of maximal comma-free codes described respectively by Eastman and by Scholtz and the notions of Hall sets and Lazard sets introduced in connection with factorizations of free monoids and bases of free Lie algebras. |
The aim of this paper is to study determinants of matrices related to the Pascal triangle. |
We show for each positive integer $a$ that, if $\mathcal{M}$ is a minor-closed class of matroids not containing all rank-$(a+1)$ uniform matroids, then there exists an integer $c$ such that either every rank-$r$ matroid in $\mathcal{M}$ can be covered by at most $r^c$ rank-$a$ sets, or $\mathcal{M}$ contains the GF$(q)... |
A bipartite graph is {\em pseudo 2--factor isomorphic} if all its 2--factors have the same parity of number of circuits. In \cite{ADJLS} we proved that the only essentially 4--edge-connected pseudo 2--factor isomorphic cubic bipartite graph of girth 4 is $K_{3,3}$, and conjectured \cite[Conjecture 3.6]{ADJLS} that the... |
The Catalan numbers occur in various counting problems in combinatorics. This paper reveals a connection between the Catalan numbers and list colouring of graphs. |
The eccentric connectivity index $\xi^c$ is a distance--based molecular structure descriptor that was recently used for mathematical modeling of biological activities of diverse nature. We prove that the broom has maximum $\xi^c$ among trees with a fixed maximum vertex degree, and characterize such trees with minimum ... |
Let S(n) be the symmetric group on n points. A subset S of S(n) is intersecting if for any pair of permutations \pi, \sigma in S there is a point i in {1,...,n} such that \pi(i)=\sigma(i). |
The purpose of this paper is to find an explicit formula and asymptotic estimate for the total number of sum of weighted records over set partitions of $[n]$ <br>in terms of Bell numbers. For that we study the generating function for the number of set partitions of $[n]$ according to the statistic sum of weighted reco... |
We present combinatorial and analytical results concerning a Sheffer sequence with a generating function of the form $G(x,z)=Q(z)^{x}Q(-z)^{1-x}$, where $Q$ is a quadratic polynomial with real zeros. By using the properties of Riordan matrices we address combinatorial properties and interpretations of our Sheffer sequ... |
Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G,\lambda)=\sum_{i=0}^{n} d(G,i) \lambda^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. |
Some identities of Chebyshev polynomials are deduced from Waring's formula on symmetric functions. In particular, these formulae generalize some recent results of Grabner and Prodinger. |
The linear system $|D|$ of a divisor $D$ on a metric graph has the structure of a cell complex. We introduce the anchor divisors and anchor cells in it - they serve as the landmarks for us to compute the f-vector of the complex and find all cells in the complex. |
Employing a recent technology of tree surgery we prove a ``deletion-constriction'' formula for products of rooted spanning trees on weighted directed graphs that generalizes deletion-contraction on undirected graphs. The formula implies that, letting $\tau_x^\varnothing$, $\tau_x^+$, and $\tau_x^-$ be the root... |
We study optimal configurations of Queens on a square chessboard, defined as those covering the maximum number of squares. For a fixed number of Queens, $q$, we prove the existence of two thresholds in board size: a non-attacking threshold beyond which all optimal configurations are pairwise non-attacking, and a stabi... |
In 1967, Grünmbaum conjectured that any $d$-dimensional polytope with $d+s\leq 2d$ vertices has at least <br>\[\phi_k(d+s,d) = {d+1 \choose k+1 }+{d \choose k+1 }-{d+1-s \choose k+1 } \] $k$-faces. This conjecture along with the characterization of equality cases was recently proved by the author. |
We study the existence of a directed Hamilton cycle in random digraphs with $m$ edges where we condition on minimum in- and out-degree at least one. Denote such a random graph by $D_{n,m}^{(\delta\geq1)}$. |
We consider a family of card shuffles of $n$ cards, where the allowed moves involve transpositions corresponding to the Jucys--Murphy elements of $\{S_m\}_{m \leq n}$. We diagonalize the transition matrix of these shuffles. |
The $k$th power of a graph $G$, denoted $G^k$, has the same vertex set as $G$, and two vertices are adjacent in $G^k$ if and only if there exists a path between them in $G$ of length at most $k$. A $K_r$-factor in a graph is a spanning subgraph in which every component is a complete graph of order $r$. |
The boxicity (respectively cubicity) of a graph $G$ is the minimum non-negative integer $k$, such that $G$ can be represented as an intersection graph of axis-parallel $k$-dimensional boxes (respectively $k$-dimensional unit cubes) and is denoted by $box(G)$ (respectively $cub(G)$). It was shown by Adiga and Chandran ... |
Let $H$ and $G$ be graphs on $n$ vertices, where $n$ is sufficiently large. We prove that if $H$ has Ore-degree at most 5 and $G$ has minimum degree at least $2n/3$ then $H\subset G. |
Dorpalen-Barry et al. proved Elser's conjecture about sign of Elser's number by interpreting them as certain sums of reduced Euler characteristics of an abstract simplicial complex known as $U$-nucleus complex. We prove a conjecture posed by them regarding the homology of $U$-nucleus complex. |
We prove that the Tutte polynomial of a coloopless paving matroid is convex along the portions of the line segments x+y=p lying in the positive quadrant. Every coloopless paving matroids is in the class of matroids which contain two disjoint bases or whose ground set is the union of two bases of M*. |
We give some results and conjectures about recurrence relations for certain sequences of binomial sums. |
Hyper-Wiener index was introduced as one of the main generalizations of the well known Wiener index. Through the years properties of the Wiener index have been extensively studied in both Mathematics and Chemistry. |
This article illustrates the dynamical concept of $homomesy$ in three kinds of dynamical systems -- combinatorial, piecewise-linear, and birational -- and shows the relationship between these three settings. In particular, we show how the rowmotion and promotion operations of Striker and Williams can be lifted to (con... |
We express the matroid polytope $P_M$ of a matroid $M$ as a signed Minkowski sum of simplices, and obtain a formula for the volume of $P_M$. This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian $Gr_{k,n}$. |
An ordinal-valued metric taking its values in the set of all countable ordinals can be assigned to a metrizable set of nodes in a transfinite graph. Then, a variety of results concerning nodal eccentricities, radii, diameters, centers, peripheries, and blocks can be extended to transfinite graphs. |
We introduce the epidemic quasimetric on graphs and study its behavior with respect to clustering techniques. In particular we compare its behavior to known objects such as the graph distance, effective resistance, and modulus of path families. |
In this paper, we obtain a Schwartz-Zippel type estimate for homogenous finite field polynomials. Specifically, we use a probabilistic recursion technique to find upper and lower bounds for the number of zeros of a homogenous polynomial and illustrate our result with two examples involving perfect matching in bipartit... |
The Directed Oberwolfach Problem can be considered as the directed version of the well-known Oberwolfach Problem, first mentioned by Ringel at a conference in Oberwolfach, Germany in 1967. In this paper, we describe some new partial results on the Directed Oberwolfach Problem with variable cycle lengths. |
Van der Holst and Pendavingh introduced a graph parameter $\sigma$, which coincides with the more famous Colin de Verdière graph parameter $\mu$ for small values. However, the definition of $\sigma$ is much more geometric/topological directly reflecting embeddability properties of the graph. |
We show that the set of binary words containing overlaps is not unambiguously context-free and that the set of ternary words containing overlaps is not context-free. We also show that the set of binary words that are not subwords of the Thue-Morse word is not unambiguously context-free. |
We are interested in overgroups of the automorphism group of the Rado graph. One class of such overgroups is completely understood; this is the class of reducts. |
Lambda words are sequences obtained by encoding the differences between ordered elements of the form i+j\theta, where i and j are non-negative integers and 1 < \theta <2. Lambda words are right-infinite words defined over an infinite alphabet that have connections with Sturmian words, Christoffel words, and inte... |
The disjoint paths problem is a fundamental problem in algorithmic graph theory and combinatorial optimization. For a given graph $G$ and a set of $k$ pairs of terminals in $G$, it asks for the existence of $k$ vertex-disjoint paths connecting each pair of terminals. |
Lattice geometries and random walks on them are of great interest for their applications in different fields such as physics, chemistry, and computer science. In this work, we focus on multi-headed lattices and study properties of the Green functions for these lattices such as the associated differential equations and... |
The monopole-dimer model introduced recently is an exactly-solvable signed generalisation of the dimer model. We show that the partition function of the monopole-dimer model on a graph invariant under a fixed-point free involution is a perfect square. |
Bicliques are complements of bipartite graphs; as such each consists of two cliques joined by a number of edges. In this paper we study algebraic aspects of the chromatic polynomials of these graphs. |
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