Related papers: A note on a Broken-cycle Theorem for hypergraphs
In this article we consider certain well-known polynomials associated with graphs including the independence polynomial and the chromatic polynomial. These polynomials count certain objects in graphs: independent sets in the case of the…
We consider a refinement of the partition function of graph homomorphisms and present a quasi-polynomial algorithm to compute it in a certain domain. As a corollary, we obtain quasi-polynomial algorithms for computing partition functions…
The independence polynomial of a hypergraph is the generating function for its independent (vertex) sets with respect to their cardinality. This article aims to discuss several recurrence relations for the independence polynomial using some…
Given a graph $G$, the $k$-coloring graph $\mathcal{C}_k(G)$ is constructed by selecting proper $k$-colorings of $G$ as vertices, with an edge between two colorings if they differ in the color of exactly one vertex. The number of vertices…
The bivariate chromatic polynomial $\chi_G(x,y)$ of a graph $G = (V, E)$, introduced by Dohmen-P\"{o}nitz-Tittmann (2003), counts all $x$-colorings of $G$ such that adjacent vertices get different colors if they are $\le y$. We extend this…
The dicycle transversal number t(D) of a digraph D is the minimum size of a dicycle transversal of D, i. e. a set T of vertices of D such that D-T is acyclic. We study the following problem: Given a digraph D, decide if there is a dicycle B…
The reliability polynomial of a graph gives the probability that a graph remains operational when all its edges could fail independently with a certain fixed probability. In general, the problem of finding uniformly most reliable graphs…
Dirac proved that each $n$-vertex $2$-connected graph with minimum degree at least $k$ contains a cycle of length at least $\min\{2k, n\}$. We consider a hypergraph version of this result. A Berge cycle in a hypergraph is an alternating…
We describe the topology of a general polynomial mapping $F=(f, g):X\to\Bbb C^2$, where $X$ is a complex plane or a complex sphere.
Let $\Delta(G)$ be the maximum degree of a graph $G$. Brooks' theorem states that the only connected graphs with chromatic number $\chi(G)=\Delta(G)+1$ are complete graphs and odd cycles. We prove a fractional analogue of Brooks' theorem in…
Answering a question of Kalai and Meshulam, we prove that graphs without induced cycles of length $3k$ have bounded chromatic number. This implies the very first case of a much broader question asserting that every graph with large…
The transmission of a connected hypergraph is defined as the summation of distances between all unordered pairs of distinct vertices. We determine the unique uniform unicyclic hypergraphs of fixed size with minimum and maximum…
A Hamiltonian path (cycle) in a graph is a path (cycle, respectively) which passes through all of its vertices. The problems of deciding the existence of a Hamiltonian cycle (path) in an input graph are well known to be NP-complete, and…
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 conjecture that a 2-connected graph $G$ of order $n$, in which $d(x)+d(y)\geq n-k$ for every pair of non-adjacent vertices $x$ and $y$, contains a cycle of length $n-k$ ($k<n/2$), unless $G$ is bipartite and $n-k$ is odd. This…
This report provides an overview of theorems and statements related to a conjecture stated by D.W. Barnette in 1969 (which is an open problem in graph theory): Every cubic, bipartite, polyhedral graph contains a Hamilton cycle.
We generalize the Harary-Sachs theorem to $k$-uniform hypergraphs: the codegree-$d$ coefficient of the characteristic polynomial of a uniform hypergraph ${\cal H}$ can be expressed as a weighted sum of subgraph counts over certain…
Let $X$ be a smooth projective variety defined over a finite field. We show that any algebraic $1$-cycle on $X$ is rationally equivalent to a smooth $1$-cycle, which is a $\mathbb{Z}$-linear combination of smooth curves on $X$. We also…
We show in this paper that a split-comparability graph $G$ has chromatic index equal to $\Delta(G) + 1$ if and only if $G$ is neighborhood-overfull. That implies the validity of the Overfull Conjecture for the class of split-comparability…
Geometric representations of cycles in quandle homology theory are given in terms of colored knot diagrams. Abstract knot diagrams are generalized to diagrams with exceptional points which, when colored, correspond to degenerate cycles.…