Searcharxiv⌕ Search

arXiv subjects

Elliot Krop

Publications and source records attributed to Elliot Krop.

At least 19 recordsLinked to original sources
arXiv2026

Extremal Chowla sets and their linear analogues: A human-AI mathematical investigation using Co-Scientist

We introduce an extremal invariant associated with Chowla-type order conditions in finite groups. A nonempty subset $S$ of a finite group $G$ is called a Chowla set if every element of $S$ has order greater than $|S|$, and we write $C(G)$ for the maximum cardinality of such a set. We first show that $C(G)$ is determined by the distribution of element orders in $G$. For cyclic groups, we derive an exact divisor formula and characterize the integers $n$ for which $C(\mathbb{Z}/n\mathbb{Z})=\varphi(n)$. We prove that $\liminf_{n\to\infty}C(\mathbb{Z}/n\mathbb{Z})/\varphi(n)=1$, whereas $\limsup_{n\to\infty}C(\mathbb{Z}/n\mathbb{Z})/\varphi(n)=\infty$, and we determine the corresponding lower and upper limits under normalization by $n$. For finite abelian groups, we obtain an explicit formula in terms of the invariant-factor decomposition, together with a closed formula for finite abelian $p$-groups. We then develop a linear analogue for finite field extensions. A nonzero $K$-subspace $A$ of an extension $L/K$ is called a Chowla subspace if $[K(a):K]>\dim_K A$ for every nonzero $a\in A$. Since this condition depends on $\dim_K A$, it does not generally require every nonzero element of $A$ to generate $L$ over $K$. Nevertheless, when $L/K$ is finite and separable, we prove the exact formula $C(L/K)=[L:K]-d_{\max}(L/K)$, where $d_{\max}(L/K)$ is the largest degree over $K$ of a proper intermediate field. For finite fields, we give a direct proof in every degree using a normal-basis construction. This work was developed through an expert-guided human-AI collaboration. A reasoning-focused configuration of Co-Scientist was used to explore examples and potential proof strategies. The authors formulated the problem, independently verified and completed all arguments, and wrote the final proofs.

math.NT↗
arXiv2026

Odd-cycle defects in the Alon-Friedland bound

Let \(G\) be a finite simple graph. We study perfect matchings through two complementary viewpoints: reductions to bipartite permanent terms and the directed cycle covers counted by the ordinary adjacency permanent. The main identity is an explicit odd-cycle-indexed form of the classical cycle-cover expansion: it separates \(\perfmat(G)^2\), the even-cycle-cover contribution coming from superposing two perfect matchings, from the contribution of cycle covers containing odd cycles. This gives a nonnegative odd-cycle defect \(\delta_{\mathrm{odd}}(G)\). Combining the identity with the Bregman--Minc inequality yields a structural refinement of the Alon--Friedland degree-sequence bound. We prove product, positivity, and fractional-perfect-matching interpretations for the defect; show that for \(K_{2n}\) the defect asymptotically accounts for almost the entire Bregman--Minc target; and derive from this a derangement identity. We also study near equality in the Alon--Friedland bound: we classify all one-edge perturbations of the extremal graphs, record the uniform obstruction coming from \(K_4\), and formulate a sharp bounded-degree gap problem whose two natural candidate extremal graphs cross between maximum degrees \(9\) and \(10\).

math.CO↗
arXiv2026

A $2/3$ Bound for Vizing's Conjecture

Vizing's conjecture, dating back to 1963, asserts that \[ \gamma(G\mathbin{\square}H) \geq \gamma(G)\gamma(H) \] for all finite graphs $G$ and $H$, where $\gamma$ denotes the domination number and $\square$ denotes the Cartesian product. In 2000, Clark and Suen proved the universal bound \[ \gamma(G\mathbin{\square}H) \geq \frac{1}{2}\gamma(G)\gamma(H). \] Recently, Steiner obtained the first constant-factor improvement of the Clark--Suen bound, proving that \[ \gamma(G\mathbin{\square}H) \geq \frac{5+\sqrt{73}}{24}\gamma(G)\gamma(H) \approx 0.5643\,\gamma(G)\gamma(H). \] In this paper, we further improve the universal constant by proving that \[ \gamma(G\mathbin{\square}H) \geq \frac{2}{3}\gamma(G)\gamma(H) \] for all finite graphs $G$ and $H$. Thus, we raise the best known universal constant in the approximate form of Vizing's conjecture from $(5+\sqrt{73})/24$ to $2/3$.

math.CO↗
arXiv2025

Matchings in Matroids over Abelian Groups, III

This paper develops matroidal analogues of classical results on matchings in abelian groups. By embedding matroid ground sets in an abelian group, we introduce base matchings between matroid bases, recover the group-theoretic setting in the uniform matroid case, and derive structural and combinatorial criteria for their existence. Our main focus is on paving matroids. We prove self-matchability for paving matroids, extend asymmetric matchability results using the hyperplane-nullity parameter, and show that stressed hyperplanes provide a natural route to matchability through relaxation.

math.CO↗
arXiv2024

The Cordiality Game and the Game Cordiality Number

The cordiality game is played on a graph $G$ by two players, Admirable (A) and Impish (I), who take turns selecting \track{unlabeled} vertices of $G$. Admirable labels the selected vertices by $0$ and Impish by $1$, and the resulting label on any edge is the sum modulo $2$ of the labels of the vertices incident to that edge. The two players have opposite goals: Admirable attempts to minimize the number of edges with different labels as much as possible while Impish attempts to maximize this number. When both Admirable and Impish play their optimal games, we define the \emph{game cordiality number}, $c_g(G)$, as the absolute difference between the number of edges labeled zero and one. Let $P_n$ be the path on $n$ vertices. We show $c_g(P_n)\le \frac{n-3}{3}$ when $n \equiv 0 \pmod 3$, $c_g(P_n)\le \frac{n-1}{3}$ when $n \equiv 1 \pmod 3$, and $c_g(P_n)\le \frac{n+1}{3}$ when $n \equiv 2\pmod 3$. Furthermore, we show a similar bound, $c_g(T) \leq \frac{|T|}{2}$ holds for any tree $T$.

math.CO↗
arXiv2021

Grundy domination of forests and the strong product conjecture

A maximum sequence $S$ of vertices in a graph $G$, so that every vertex in $S$ has a neighbor which is independent, or is itself independent, from all previous vertices in $S$, is called a Grundy dominating sequence. The Grundy domination number, $\gamma_{gr}(G)$, is the length of $S$. We show that for any forest $F$, $\gamma_{gr}(F)=|V(T)|-|\mathcal{P}|$ where $\mathcal{P}$ is a minimum partition of the non-isolate vertices of $F$ into caterpillars in which if two caterpillars of $\mathcal{P}$ have an edge between them in $F$, then such an edge must be incident to a non-leaf vertex in at least one of the caterpillars. We use this result to show the strong product conjecture of B. Bre\v{s}ar, Cs. Bujt\'{a}s, T. Gologranc, S. Klav\v{z}ar, G. Ko\v{s}mrlj, B. Patk\'{o}s, Zs. Tuza, and M. Vizer, Dominating sequences in grid-like and toroidal graphs, Electron. J. Combin. 23(4): P4.34 (2016), for all forests. Namely, we show that for any forest $G$ and graph $H$, $\gamma_{gr}(G \boxtimes H) = \gamma_{gr}(G) \gamma_{gr}(H)$. We also show that every connected graph $G$ has a spanning tree $T$ so that $\gamma_{gr}(G)\le \gamma_{gr}(T)$ and that every non-complete connected graph contains a Grundy dominating set $S$ so that the induced subgraph of $S$ contains no isolated vertices.

math.CO↗
arXiv2019

Upper bounds for inverse domination in graphs

In any graph $G$, the domination number $\gamma(G)$ is at most the independence number $\alpha(G)$. The Inverse Domination Conjecture says that, in any isolate-free $G$, there exists pair of vertex-disjoint dominating sets $D, D'$ with $|D|=\gamma(G)$ and $|D'| \leq \alpha(G)$. Here we prove that this statement is true if the upper bound $\alpha(G)$ is replaced by $\frac{3}{2}\alpha(G) - 1$ (and $G$ is not a clique). We also prove that the conjecture holds whenever $\gamma(G)\leq 5$ or $|V(G)|\leq 16$.

math.CO↗
arXiv2018

A Vizing-type result for semi-total domination

A set of vertices $S$ in a simple isolate-free graph $G$ is a semi-total dominating set of $G$ if it is a dominating set of $G$ and every vertex of $S$ is within distance 2 or less with another vertex of $S$. The semi-total domination number of $G$, denoted by $\gamma_{t2}(G)$, is the minimum cardinality of a semi-total dominating set of $G$. In this paper, we study semi-total domination of Cartesian products of graphs. Our main result establishes that for any graphs $G$ and $H$, $\gamma_{t2}(G\,\square\, H)\ge \frac{1}{3}\gamma_{t2}(G)\gamma_{t2}(H)$.

math.CO↗
arXiv2017

On a Vizing-type integer domination conjecture

Given a simple graph $G$, a dominating set in $G$ is a set of vertices $S$ such that every vertex not in $S$ has a neighbor in $S$. Denote the domination number, which is the size of any minimum dominating set of $G$, by $\gamma(G)$. For any integer $k\ge 1$, a function $f : V (G) \rightarrow \{0, 1, . . ., k\}$ is called a \emph{$\{k\}$-dominating function} if the sum of its function values over any closed neighborhood is at least $k$. The weight of a $\{k\}$-dominating function is the sum of its values over all the vertices. The $\{k\}$-domination number of $G$, $\gamma_{\{k\}}(G)$, is defined to be the minimum weight taken over all $\{k\}$-domination functions. Bre\v{s}ar, Henning, and Klav\v{z}ar (On integer domination in graphs and Vizing-like problems. \emph{Taiwanese J. Math.} {10(5)} (2006) pp. 1317--1328) asked whether there exists an integer $k\ge 2$ so that $\gamma_{\{k\}}(G\square H)\ge \gamma(G)\gamma(H)$. In this note we use the Roman $\{2\}$-domination number, $\gamma_{R2}$ of Chellali, Haynes, Hedetniemi, and McRae, (Roman $\{2\}$-domination. \emph{Discrete Applied Mathematics} {204} (2016) pp. 22-28.) to prove that if $G$ is a claw-free graph and $H$ is an arbitrary graph, then $\gamma_{\{2\}}(G\square H)\ge \gamma_{R2}(G\square H)\ge \gamma(G)\gamma(H)$, which also implies the conjecture for all $k\ge 2$.

math.CO↗
arXiv2017

Vizing-type bounds for graphs with induced subgraph restrictions

For any graphs $G$ and $H$, we say that a bound is of Vizing-type if $\gamma(G\square H)\geq c \gamma(G)\gamma(H)$ for some constant $c$. We show several bounds of Vizing-type for graphs $G$ with forbidden induced subgraphs. In particular, if $G$ is a triangle and $K_{1,r}$-free graph, then for any graph $H$, $\gamma(G\square H)\geq \frac{r}{2r-1}\gamma(G)\gamma(H)$. If $G$ is a $K_r$ and $P_5$-free graph for some integer $r\geq 2$, then for any graph $H$, $\gamma(G\square H)\geq \frac{r-1}{2r-3}\gamma(G)\gamma(H)$. We do this by bounding the power of $G$, $\pi(G)$. We show that if $G$ is claw-free and $P_6$-free or $K_4$ and $P_5$-free, then for any graph $H$, $\gamma(G\square H)\geq \gamma(G)\gamma(H)$. Furthermore, we show Vizing-type bounds in terms of the diameter of $G$.

math.CO↗
arXiv2016

Vizing's conjecture for cographs

We show that if $G$ is a cograph, that is $P_4$-free, then for any graph $H$, $\gamma(G\square H)\geq \gamma(G)\gamma(H)$. By the characterization of cographs as a finite sequence of unions and joins of $K_1$, this result easily follows from that of Bartsalkin and German. However, the techniques used are new and may be useful to prove other results.

math.CO↗
arXiv2016

A new bound for Vizing's conjecture

For any graph $G$, we define the power $\pi(G)$ as the minimum of the largest number of neighbors in a $\gamma$-set of $G$, of any vertex, taken over all $\gamma$-sets of $G$. We show that $\gamma(G\square H)\geq \frac{\pi(G)}{2\pi(G) -1}\gamma(G)\gamma(H)$. Our methods allow us to prove the following statements for any graphs $G$ and $H$, (1) $\gamma(G\square H)\geq \frac{\lceil \frac{\gamma (G)}{2}\rceil}{2\lceil \frac{\gamma (G)}{2}\rceil-1}\gamma(G)\gamma(H)$ for odd $\gamma(G)$, (2) $\gamma(G\square H)\geq \frac{\gamma (G)}{2\gamma (G)-2}\gamma(G)\gamma(H)$, for even $\gamma(G)$, and (3) a short proof of Vizing's conjecture where $\gamma(G)=3$. Our argument relies on establishing efficient correspondences between dominating vertices and subsets of their neighborhoods and then showing a sufficient number of dominating vertices that horizontally dominate vertically undominated cells.

math.CO↗
arXiv2016

All trees are six-cordial

For any integer $k>0$, a tree $T$ is $k$-cordial if there exists a labeling of the vertices of $T$ by $\mathbb{Z}_k$, inducing a labeling on the edges with edge-weights found by summing the labels on vertices incident to a given edge modulo $k$ so that each label appears on at most one more vertex than any other and each edge-weight appears on at most one more edge than any other. We prove that all trees are six-cordial by an adjustment of the test proposed by Hovey (1991) to show all trees are $k$-cordial.

math.CO↗
arXiv2015

A class of graphs approaching Vizing's conjecture

For any graph $G=(V,E)$, a subset $S\subseteq V$ \emph{dominates} $G$ if all vertices are contained in the closed neighborhood of $S$, that is $N[S]=V$. The minimum cardinality over all such $S$ is called the domination number, written $\gamma(G)$. In 1963, V.G. Vizing conjectured that $\gamma(G \square H) \geq \gamma(G)\gamma(H)$ where $\square$ stands for the Cartesian product of graphs. In this note, we define classes of graphs $\mathcal{A}_n$, for $n\geq 0$, so that every graph belongs to some such class, and $\mathcal{A}_0$ corresponds to class $A$ of Bartsalkin and German. We prove that for any graph $G$ in class $\mathcal{A}_1$, $\gamma(G\square H)\geq \left(\gamma(G)-\sqrt{\gamma(G)}\right)\gamma(H)$.

math.CO↗
arXiv2015

Vizing's Conjecture for Almost All Pairs of Graphs

For any graph $G=(V,E)$, a subset $S\subseteq V$ $dominates$ $G$ if all vertices are contained in the closed neighborhood of $S$, that is $N[S]=V$. The minimum cardinality over all such $S$ is called the domination number, written $γ(G)$. In 1963, V.G. Vizing conjectured that $γ(G \square H) \geq γ(G)γ(H)$ where $\square$ stands for the Cartesian product of graphs. In this note, we prove that if $\left|G\right|\geq γ(G)γ(H)$ and $\left|H\right|\geq γ(G)γ(H)$, then the conjecture holds. This result quickly implies Vizing's conjecture for almost all pairs of graphs $G,H$ with $\left|G\right|\geq \left|H\right|$, satisfying $\left|G\right|\leq q^{\frac{\left|H\right|}{\log_q\left|H\right|}}$ for $q=\frac{1}{1-p}$ and $p$ the edge probability of the Erdős-Rényi random graph.

math.CO↗
arXiv2014

A counterexample to a conjecture of Ghosh

We answer two questions of Shamik Ghosh in the negative. We show that there exists a lobster tree of diameter less than 6 which accepts no alpha-labeling with two central vertices labeled by the critical number and the maximum vertex label. We also show a simple example of a tree of diameter 4, with an even degree central vertex which does not accept a maximum label in any graceful labeling.

math.CO↗
arXiv2014

On small Mixed Pattern Ramsey numbers

We call the minimum order of any complete graph so that for any coloring of the edges by $k$ colors it is impossible to avoid a monochromatic or rainbow triangle, a Mixed Ramsey number. For any graph $H$ with edges colored from the above set of $k$ colors, if we consider the condition of excluding $H$ in the above definition, we produce a \emph{Mixed Pattern Ramsey number}, denoted $M_k(H)$. We determine this function in terms of $k$ for all colored $4$-cycles and all colored $4$-cliques. We also find bounds for $M_k(H)$ when $H$ is a monochromatic odd cycles, or a star for sufficiently large $k$. We state several open questions.

math.CO↗