Mathematics Problem Archive

Given a symmetric crossing supermodular function $p:2^V\to \mathbb{R}$ and positive integers $n_1,n_2,\dots,n_k$, does there exist a hypergraph H=(V,E...

Let M be a matroid on ground set S, and suppose that S can be partitioned into k bases. Is it true that there is a cyclic ordering of the elements of ...

What is the complexity of deciding kernel-perfectness in various classes of digraphs?...

Let G=(V,E) be an undirected graph, and $r \in V$ a root node. G is called rooted (k,l)-connected if G-X is $(k-\vert X\vert)l$-edge-connected for any...

Let D=(V,A) be a digraph whose underlying graph is k-partition-connected, and let $r_0 \in V$ be a node of in-degree 0. Suppose that the in-degree of ...

Let G be a graph that is rigid in two-dimensional space. Can we determine in polynomial time the minimum number of edges whose deletion from G results...

Does there exist a value k so that in every k-arc-connected directed graph D=(V,A) and for every node $v\in V$, there is a spanning in-arborescence an...

Let f be a 2-polymatroid function on S that has a matroid representation $M=(S \times \{1,2\},r)$ with the following property: $|C\cap \{(e,1),(e,2)\}...

In a graph G=(V,E) with a root node r, two spanning trees $T_1$ and $T_2$ are called edge-independent if for any node x in V-r, the unique paths betwe...

Is it true that every graph G is equitably k-list-colourable for any $k \geq \Delta(G)+1$?...

Let A be an $n \times n$ 0-1 matrix, and suppose that the facets of the polyhedron $P=\{x: A x \leq {\mathbf 1},\ x \leq {\mathbf 1}\}$ are coloured b...

In which classes of digraphs can we decide if a kernel exists and find one in polynomial time?...

How does gonality change if each edge of the graph is subdivided $k$ times?...

An orientation of a hypergraph is a directed hypergraph obtained by choosing a single head-node in each hyperedge. We call a set of orientations of a ...

Is it true that if an undirected graph G with terminal set T is 2k-element-connected, then it has a k-element-connected orientation?...

Given a digraph D=(V+s,A) which is k-arc-connected in V, what is the maximum number of (disjoint) pairs of arcs, consisting of entering and leaving ar...

Let D=(V,A) be an acyclic digraph with designated root-nodes $r_1,...,r_k\in V$. Let $U_1,...,U_k$ be convex node sets with $r_i\in U_i$. Is it true t...

For a digraph $D=(V,A)$, we call a nonempty $C\subseteq A$ a dicut if there is some $X\subseteq V$ such that no edge enters $X$ and $C$ consists of th...

Given some matroids on the same ground set $S$, a colouring of $S$ is called proper if each monochromatic set is independent in each matroid. Let $M_1...

We are given a hypergraph $G_0=(V,\mathcal{E}_0)$ of rank at most $k$ (where $k$ is fixed, not part of the input) and a symmetric function $r:V\times ...

Let D=(V,E) be a directed graph that is the union of two disjoint directed spanning trees. Can we characterize when does D have a directed spanning tr...

Let (G,<) be a preference system with ties where in every odd cycle there is a node that prefers its clockwise neighbour to its other neighbour, and t...

Is it true that if all vertices of a base polyhedron B are in $\{0,1,-1\}^n$ and $0 \in B$, then B has a vertex v such that -v is also a vertex?...

Let $\mathcal{C}$ be a clutter on ground set V, and let $\mathcal{B}$ be its blocker. Is it true that $\mathcal{C}$ is nonideal if and only if there e...

Let G=(V,E) be a mixed graph with non-negative edge-lengths and let $s,t \in V$. Can we find in polynomial time an orientation where the sum of the le...

Given a digraph D=(V,A) and non-negative even-valued arc weights $w_a\ (a \in A)$, can we find in polynomial time a w-vertex cover $x$ of the underlyi...

Let P be the set of partitions of a ground set S. We allow two operations on P: (1) splitting a class into two arbitrary classes and (2) joining two c...

Let G=(V,E) be a bipartite graph, and $c_1,\dots c_k$ positive reals whose sum is 1. Can E always be partitioned into k parts $E_1,\dots,E_k,$ so that...

For which classes of digraphs can we explicitly give a linear description of the convex hull of kernels?...

A quasi-kernel of a digraph $D$ is an independent vertex set $K$ sucht that every vertex is reachable from $K$ in $D$ by a path of length at most two....

Given k disjoint matchings in a bipartite graph, a rainbow matching is a matching that contains one edge from each of them. Is it true that any family...

Given a crossing negamodular function $R:2^V\to \mathbb{Z}$ such that $R(X)\ne 1$ for every $X\subseteq V$ and a hypergraph $G_0=(V,\mathcal{E}_0)$, f...

A graph G is said to be a Seymour graph if for any edge set F that satisfies $|C\cap F|\le |C\setminus F|$ for every circuit C of G, there exist $|F|$...

Let M be a matroid, and let A and B be two bases of M. A subset X of A and a subset Y of B, both of size k, form a serial symmetric exchange with resp...

Let $p_1$ and $p_2$ be integer skew-supermodular set functions on ground set S such that $\max\{p_1(X),p_2(X)\}\leq \min\{|X|,k\}$ for every $X \subse...

Let G=(V,E) be a graph with maximum degree $\Delta \geq 2$, and let M=(V,r) be a matroid that has $2 \Delta$ disjoint bases. Is it true that M has $2 ...

For a loopless matroid $M=(S,r)$, let $\Delta(M)=\max_{X\subseteq S} |X|/r(X)$. Let $M_1=(S,r_1)$ and $M_2=(S,r_2)$ be two arbitrary loopless matroids...

Graham: A set of points S is Euclidean Ramsey if, for every k, there exists an N so that every k-coloring of Euclidean N-space contains a monochromati...

Graham: For every non-equilateral triangle T, show that it is possible to color the plane with three colors so that there is no monochromatic (congrue...

Pach : Suppose a geometric graph has no pairwise k-crossing lines. That is, no k edges all cross each other. Must the graph have O_(k)(n) edges?...

: Suppose we begin with a set of points S in the plane. Let T(S) be the set of points one gets by taking all lines through pairs of points in S, and t...

Solymosi : Suppose a family V of n points are chosen in R^(3) and L is a collection of lines so that every triangle spanned by three points of V is pi...

What is the minimum number of n-simplexes needed to triangulate the n-cube? See this....

How many congruent regular tetrahedra can touch at a point? Easy to show it's at least 20, and at most 22. Apparently, this has been open a long time....

Straus: Is every polygonal room in the plane illuminable from some point? See this....

Erdős: Does lim R(k,k)^(1/k) exist? What is it? (If it exists, it's between sqrt(2) and 4.) See "Small Ramsey Numbers" by Stanislaw Radziszowski....

Pach, Tóth: Define the "crossing number" of a graph to be the minimum number of (topological) crossings of edges in any straight-line embedding in the...

Erdős: Show that every (1/2+ľ)|E(ő_(n))| edges of the n-cube ő_(n) contains a C_(4) when n is sufficiently large. (The best known value of ľ is around...

Graham: Suppose that G is a tree. Denote by L(G) the line graph of G. Is the sequence |G|, |L(G)|, |L(L(G))|, |L(L(L(G)))| ... unique to G? That is, c...

: What is the list-chromatic number of Sudoku? That is, suppose one places k symbols (aka colors) in each cell of a Sudoku board -- not necessarily al...