Singmaster's conjecture ★★

Author(s): Singmaster

Conjecture   There is a finite upper bound on the multiplicities of entries in Pascal's triangle, other than the number $ 1 $.

The number $ 2 $ appears once in Pascal's triangle, $ 3 $ appears twice, $ 6 $ appears three times, and $ 10 $ appears $ 4 $ times. There are infinite families of numbers known to appear $ 6 $ times. The only number known to appear $ 8 $ times is $ 3003 $. It is not known whether any number appears more than $ 8 $ times. The conjectured upper bound could be $ 8 $; Singmaster thought it might be $ 10 $ or $ 12 $. See Singmaster's conjecture.

Keywords: Pascal's triangle

Waring rank of determinant ★★

Author(s): Teitler

Question   What is the Waring rank of the determinant of a $ d \times d $ generic matrix?

For simplicity say we work over the complex numbers. The $ d \times d $ generic matrix is the matrix with entries $ x_{i,j} $ for $ 1 \leq i,j \leq d $. Its determinant is a homogeneous form of degree $ d $, in $ d^2 $ variables. If $ F $ is a homogeneous form of degree $ d $, a power sum expression for $ F $ is an expression of the form $ F = \ell_1^d+\dotsb+\ell_r^d $, the $ \ell_i $ (homogeneous) linear forms. The Waring rank of $ F $ is the least number of terms $ r $ in any power sum expression for $ F $. For example, the expression $ xy = \frac{1}{4}(x+y)^2 - \frac{1}{4}(x-y)^2 $ means that $ xy $ has Waring rank $ 2 $ (it can't be less than $ 2 $, as $ xy \neq \ell_1^2 $).

The $ 2 \times 2 $ generic determinant $ x_{1,1}x_{2,2}-x_{1,2}x_{2,1} $ (or $ ad-bc $) has Waring rank $ 4 $. The Waring rank of the $ 3 \times 3 $ generic determinant is at least $ 14 $ and no more than $ 20 $, see for instance Lower bound for ranks of invariant forms, Example 4.1. The Waring rank of the permanent is also of interest. The comparison between the determinant and permanent is potentially relevant to Valiant's "VP versus VNP" problem.

Keywords: Waring rank, determinant

Monochromatic vertex colorings inherited from Perfect Matchings ★★★

Author(s):

Conjecture   For which values of $ n $ and $ d $ are there bi-colored graphs on $ n $ vertices and $ d $ different colors with the property that all the $ d $ monochromatic colorings have unit weight, and every other coloring cancels out?

Keywords:

Cycle Double Covers Containing Predefined 2-Regular Subgraphs ★★★

Author(s): Arthur; Hoffmann-Ostenhof

Conjecture   Let $ G $ be a $ 2 $-connected cubic graph and let $ S $ be a $ 2 $-regular subgraph such that $ G-E(S) $ is connected. Then $ G $ has a cycle double cover which contains $ S $ (i.e all cycles of $ S $).

Keywords:

Monochromatic reachability in arc-colored digraphs ★★★

Author(s): Sands; Sauer; Woodrow

Conjecture   For every $ k $, there exists an integer $ f(k) $ such that if $ D $ is a digraph whose arcs are colored with $ k $ colors, then $ D $ has a $ S $ set which is the union of $ f(k) $ stables sets so that every vertex has a monochromatic path to some vertex in $ S $.

Keywords:

3-Decomposition Conjecture ★★★

Author(s): Arthur; Hoffmann-Ostenhof

Conjecture   (3-Decomposition Conjecture) Every connected cubic graph $ G $ has a decomposition into a spanning tree, a family of cycles and a matching.

Keywords: cubic graph