Combinatorics (math.CO)

Abstract: We show that for threshold graphs, the eigenvalues of the signless Laplacian matrix interlace with the degrees of the vertices. As an application, we show that the signless Brouwer conjecture holds for threshold graphs, i.e., for threshold graphs the sum of the k largest eigenvalues is bounded by the number of edges plus k + 1 choose 2.

Abstract: We define signable, generalized signable, and $Z$-signable correspondence assignments on multigraphs, which generalize good correspondence assignments as introduced by Kaul and Mudrock. DP-colorings from these classes generalize signed colorings, signed $\mathbb{Z}_p$-colorings, and signed list colorings of signed graphs. We introduce an auxiliary digraph that allows us to prove an Alon-Tarsi style theorem for DP-colorings from $Z$-signable correspondence assignments on multigraphs, and obtain three DP-coloring analogs of the Alon-Tarsi theorem as corollaries.

Abstract: We present sharp upper bounds on the volume, Mahler volume and multiplicity for Fano simplices depending on the dimension and Gorenstein index. These bounds rely on the interplay between lattice simplices and unit fraction partitions. Moreover, we present an efficient procedure for explicitly classifying Fano simplicies of any dimension and Gorenstein index and we carry out the classification up to dimension four for various Gorenstein indices.

Abstract: In an introductory paper on dollar game played on a graph, Matt Baker wrote the following: ``The total number of borrowing moves required to win the game when playing the 'borrowing binge strategy' is independent of which borrowing moves you do in which order! Note, however, that it is usually possible to win in fewer moves by employing lending moves in combination with borrowing moves. The optimal strategy when one uses both kinds of moves is not yet understood.'' In this article, we give a lower bound on the minimum number $ M_{\text{min}} $ of such moves of an optimal algorithm in terms of the number of moves $ M_0 $ of the borrowing binge strategy. Concretely, we have: $ M_{\text{min}} \geq \frac{M_0}{n-1} $ where $ n $ is the number of vertices of the graph. This bound is tight.

Abstract: The mean subtree order of a given graph $G$, denoted $\mu(G)$, is the average number of vertices in a subtree of $G$. Let $G$ be a connected graph. Chin, Gordon, MacPhee, and Vincent [J. Graph Theory, 89(4): 413-438, 2018] conjectured that if $H$ is a proper spanning supergraph of $G$, then $\mu(H) > \mu(G)$. Cameron and Mol [J. Graph Theory, 96(3): 403-413, 2021] disproved this conjecture by showing that there are infinitely many pairs of graphs $H$ and $G$ with $H\supset G$, $V(H)=V(G)$ and $|E(H)|= |E(G)|+1$ such that $\mu(H) < \mu(G)$. They also conjectured that for every positive integer $k$, there exists a pair of graphs $G$ and $H$ with $H\supset G$, $V(H)=V(G)$ and $|E(H)| = |E(G)| +k$ such that $\mu(H) < \mu(G)$. Furthermore, they proposed that $\mu(K_m+nK_1) < \mu(K_{m, n})$ provided $n\gg m$. In this note, we confirm these two conjectures.

Abstract: Recent literature posed the problem of characterising the graph degree sequences with exactly one $3$-polytopal (i.e. planar, $3$-connected) realisation. This seems to be a difficult problem in full generality. In this paper, we characterise the sequences with exactly one self-dual $3$-polytopal realisation. An algorithm in the literature constructs a self-dual $3$-polytope for any admissible degree sequence. To do so, it performs operations on the radial graph, so that the corresponding $3$-polytope and its dual are modified in exactly the same way. To settle our question and construct the relevant graphs, we apply this algorithm, we introduce some modifications of it, and we also devise new ones. The speed of these algorithms is linear in the graph order.

Abstract: A classical result in additive combinatorics, which is a combination of Balog-Szemer\'edi-Gowers theorem and a variant of Freiman's theorem due to Ruzsa, says that if a subset $A$ of $\mathbb{F}_p^n$ contains at least $c |A|^3$ additive quadruples, then there exists a subspace $V$, comparable in size to $A$, such that $|A \cap V| \geq \Omega_c(|A|)$. Motivated by the fact that higher order approximate algebraic structures play an important role in the theory of uniformity norms, it would be of interest to find higher order analogues of the mentioned result. In this paper, we study a quadratic version of the approximate property in question, namely what it means for a set to be an approximate quadratic variety. It turns out that information on the number of additive cubes, which are 8-tuples of the form $(x, x+ a,$ $x+ b, x+ c,$ $x+ a + b, x+ a + c,$ $x+ b + c, x+ a + b + c)$, in a set is insufficient on its own to guarantee quadratic structure, and it is necessary to restrict linear structure in a given set, which is a natural assumption in this context. With this in mind, we say that a subset $V$ of a finite vector space $G$ is a $(c_0, \delta, \varepsilon)$-approximate quadratic variety if $|V| = \delta |G|$, $\|1_V - \delta\|_{\mathsf{U}^2} \leq \varepsilon$ and $V$ contains at least $c_0\delta^7 |G|^4$ additive cubes. Our main result is the structure theorem for approximate quadratic varieties, stating that such a set has a large intersection with an exact quadratic variety of comparable size.