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Register a new userAn update on a few permanent conjectures
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Fuzhen Zhang
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We review and update on a few conjectures concerning matrix permanent that are easily stated, understood, and accessible to general math audience. They are: Soules permanent-on-top conjecture${}^dagger$, Lieb permanent dominance conjecture, Bapat and Sunder conjecture${}^dagger$ on Hadamard product and diagonal entries, Chollet conjecture on Hadamard product, Marcus conjecture on permanent of permanents, and several other conjectures. Some of these conjectures are recently settled; some are still open. We also raise a few new questions for future study. (${}^dagger$conjectures have been recently settled negatively.)
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Leveraging recent advances in additive combinatorics, we exhibit explicit matrices satisfying the Restricted Isometry Property with better parameters. Namely, for $varepsilon=3.26cdot 10^{-7}$, large $k$ and $k^{2-varepsilon} le Nle k^{2+varepsilon}$, we construct $n times N$ RIP matrices of order $k$ with $k = Omega( n^{1/2+varepsilon/4} )$.
The Stern poset $mathcal{S}$ is a graded infinite poset naturally associated to Sterns triangle, which was defined by Stanley analogously to Pascals triangle. Let $P_n$ denote the interval of $mathcal{S}$ from the unique element of row $0$ of Sterns triangle to the $n$-th element of row $r$ for sufficiently large $r$. For $ngeq 1$ let begin{align*} L_n(q)&=2cdotleft(sum_{k=1}^{2^n-1}A_{P_k}(q)right)+A_{P_{2^n}}(q), end{align*} where $A_{P}(q)$ represents the corresponding $P$-Eulerian polynomial. For any $ngeq 1$ Stanley conjectured that $L_n(q)$ has only real zeros and $L_{4n+1}(q)$ is divisible by $L_{2n}(q)$. In this paper we obtain a simple recurrence relation satisfied by $L_n(q)$ and affirmatively solve Stanleys conjectures. We also establish the asymptotic normality of the coefficients of $L_n(q)$.
In 1989 Kalai stated the three conjectures A, B, C of increasing strength concerning face numbers of centrally symmetric convex polytopes. The weakest conjecture, A, became known as the ``$3^d$-conjecture. It is well-known that the three conjectures hold in dimensions d leq 3. We show that in dimension 4 only conjectures A and B are valid, while conjecture C fails. Furthermore, we show that both conjectures B and C fail in all dimensions d geq 5.
For a graph $G$, let $cp(G)$ denote the minimum number of cliques of $G$ needed to cover the edges of $G$ exactly once. Similarly, let $bp_k(G)$ denote the minimum number of bicliques (i.e. complete bipartite subgraphs of $G$) needed to cover each edge of $G$ exactly $k$ times. We consider two conjectures -- one regarding the maximum possible value of $cp(G) + cp(overline{G})$ (due to de Caen, ErdH{o}s, Pullman and Wormald) and the other regarding $bp_k(K_n)$ (due to de Caen, Gregory and Pritikin). We disprove the first, obtaining improved lower and upper bounds on $max_G cp(G) + cp(overline{G})$, and we prove an asymptotic version of the second, showing that $bp_k(K_n) = (1+o(1))n$.
The 1-2-3 Conjecture, posed by Karo{n}ski, {L}uczak and Thomason, asked whether every connected graph $G$ different from $K_2$ can be 3-edge-weighted so that every two adjacent vertices of $G$ get distinct sums of incident weights. The 1-2 Conjecture states that if vertices also receive colors and the vertex color is added to the sum of its incident edges, then adjacent vertices can be distinguished using only ${ 1,2}$. In this paper we confirm 1-2 Conjecture for 3-regular graphs. Meanwhile, we show that every 3-regular graph can achieve a neighbor sum distinguishing edge coloring by using 4 colors, which answers 1-2-3 Conjecture positively.
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