Related papers: The Purity Conjecture in Type $C$
For integers $k\ge1$ and $m\ge2$, let $g(k,m)$ be the least integer $n\ge1$ such that every graph with chromatic number at least $n$ contains a $(k+1)$-connected subgraph with chromatic number at least $m$. Refining the recent result…
We prove a number of results related to the computational complexity of recognizing well-covered graphs. Let $k$ and $s$ be positive integers and let $G$ be a graph. Then $G$ is said - $\mathbf{W_k}$ if for any $k$ pairwise disjoint…
A $(v,k,t)$ packing of size $b$ is a system of $b$ subsets (blocks) of a $v$-element underlying set such that each block has $k$ elements and every $t$-set is contained in at most one block. $P(v,k,t)$ stands for the maximum possible $b$. A…
In 1952, J.H.Braun claimed to have established a formula giving a lower bound for certain partitions of sets of integers into weakly sum-free classes. However, no proof or supporting construction was published at that time. In today's…
We investigate the existence of maximal collections of mutually noncrossing $k$-element subsets of $\left\{ 1, \dots, n \right\}$ that are invariant under adding $k\pmod n$ to all indices. Our main result is that such a collection exists if…
A basic fact in algebraic graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue 1 in the normalized adjacency matrix of the graph. In particular, the graph is…
We reconsider density matrices of graphs as defined in [quant-ph/0406165]. The density matrix of a graph is the combinatorial laplacian of the graph normalized to have unit trace. We describe a simple combinatorial condition (the "degree…
Menger's theorem tells us that if $S,T$ are sets of vertices in a graph $G$, then (for $k\ge0$) either there are $k+1$ vertex-disjoint paths between $S$ and $T$, or there is a set of $k$ vertices separating $S$ and $T$. But what if we want…
The celebrated notion of important separators bounds the number of small $(S,T)$-separators in a graph which are 'farthest from $S$' in a technical sense. In this paper, we introduce a generalization of this powerful algorithmic primitive…
In \cite{Elek} we proved that the limit of a weakly convergent sequence of finite graphs can be viewed as a graphing or a continuous field of infinite graphs. Thus one can associate a type $II_1$-von Neumann algebra to such graph sequences.…
Coarse graph theory concerns finding 'coarse' analogues of graph theory theorems, replacing disjointness with being far apart. One of the most interesting open questions is to find a coarse analogue of Menger's theorem, which characterizes…
Mader conjectured that for any tree $T$ of order $m$, every $k$-connected graph $G$ with minimum degree at least $\lfloor\frac{3k}{2}\rfloor +m-1$ contains a subtree $T'\cong T$ such that $G-V(T')$ is $k$-connected. In this paper, we give a…
In this paper we state and prove ad hoc "Separation Theorems" of the so-called Smooth Commutative Algebra, the Commutative Algebra of \(\mathcal{C}^{\infty}-\)rings. These results are formally similar to the ones we find in (ordinary)…
Given a graph $G$ a set $S\subset V(G)$ is called monophonic if every vertex in $G$ lies on some induced path between two vertices in $S$. The monophonic number, $m(G)$, of $G$, which is the smallest cardinality of a monophonic set in $G$,…
We study the spectral theory of a class of piecewise centrosymmetric Jacobi operators defined on an associated family of substitution graphs. Given a finite centrosymmetric matrix viewed as a weight matrix on a finite directed path graph…
Polignac [1] conjectured that for every even natural number $2k (k\geq1)$, there exist infinitely many consecutive primes $p_n$ and $p_{n+1}$ such that $p_{n+1}-p_n=2k$. A weakened form of this conjecture states that for every $k\geq1$,…
A decomposition of a graph is a set of subgraphs whose edges partition those of $G$. The 3-decomposition conjecture posed by Hoffmann-Ostenhof in 2011 states that every connected cubic graph can be decomposed into a spanning tree, a…
There is a rich theory of existence theorems for minimizers over reflexive Sobolev spaces (ex. Eberlein-\v{S}mulian theorem). However, the existence theorems for many variational problems over non-reflexive Sobolev spaces remain…
In this paper we prove that a set of points $B$ of PG(n,2) is a minimal blocking set if and only if $<B>=PG(d,2)$ with $d$ odd and $B$ is a set of $d+2$ points of $PG(d,2)$ no $d+1$ of them in the same hyperplane. As a corollary to the…
For the ordered set $[n]$ of $n$ elements, we consider the class $\Bscr_n$ of bases $B$ of tropical Pl\"ucker functions on $2^{[n]}$ such that $B$ can be obtained by a series of mutations (flips) from the basis formed by the intervals in…