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Michael Tait

Publications and source records attributed to Michael Tait.

71 records · Page 4Linked to original sources
arXiv2017

Three conjectures in extremal spectral graph theory

We prove three conjectures regarding the maximization of spectral invariants over certain families of graphs. Our most difficult result is that the join of $P_2$ and $P_{n-2}$ is the unique graph of maximum spectral radius over all planar graphs. This was conjectured by Boots and Royle in 1991 and independently by Cao and Vince in 1993. Similarly, we prove a conjecture of Cvetković and Rowlinson from 1990 stating that the unique outerplanar graph of maximum spectral radius is the join of a vertex and $P_{n-1}$. Finally, we prove a conjecture of Aouchiche et al from 2008 stating that a pineapple graph is the unique connected graph maximizing the spectral radius minus the average degree. To prove our theorems, we use the leading eigenvector of a purported extremal graph to deduce structural properties about that graph. Using this setup, we give short proofs of several old results: Mantel's Theorem, Stanley's edge bound and extensions, the Kővari-Sós-Turán Theorem applied to $\mathrm{ex}\left(n, K_{2,t}\right)$, and a partial solution to an old problem of Erdős on making a triangle-free graph bipartite.

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arXiv2017

Degenerate Turán problems for hereditary properties

Let $H$ be a graph and $t\geq s\geq 2$ be integers. We prove that if $G$ is an $n$-vertex graph with no copy of $H$ and no induced copy of $K_{s,t}$, then $λ(G) = O\left(n^{1-1/s}\right)$ where $λ(G)$ is the spectral radius of the adjacency matrix of $G$. Our results are motivated by results of Babai, Guiduli, and Nikiforov bounding the maximum spectral radius of a graph with no copy (not necessarily induced) of $K_{s,t}$.

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arXiv2016

A Szemerédi-Trotter type theorem, sum-product estimates in finite quasifields, and related results

We prove a Szemerédi-Trotter type theorem and a sum-product estimate in the setting of finite quasifields. These estimates generalize results of the fourth author, of Garaev, and of Vu. We generalize results of Gyarmati and Sárközy on the solvability of the equations $a + b = cd$ and $ab + 1 = cd$ over a finite field. Other analogous results that are known to hold in finite fields are generalized to finite quasifields.

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arXiv2016

Degree Ramsey numbers for even cycles

Let $H\xrightarrow{s} G$ denote that any $s$-coloring of $E(H)$ contains a monochromatic $G$. The degree Ramsey number of a graph $G$, denoted by $R_Δ(G, s)$, is $\min \{Δ(H): H \xrightarrow{s} G \}$. We consider degree Ramsey numbers where $G$ is a fixed even cycle. Kinnersley, Milans, and West showed that $R_Δ(C_{2k},s) \geq 2s$, and Kang and Perarnau showed that $R_Δ(C_4, s) = Θ(s^2)$. Our main result is that $R_Δ(C_6, s) = Θ(s^{3/2})$ and $R_Δ(C_{10}, s) = Θ(s^{5/4})$. Additionally, we substantially improve the lower bound for $R_Δ(C_{2k}, s)$ for general $k$.

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arXiv2016

Spectral bounds for the $k$-independence number of a graph

In this paper, we obtain two spectral upper bounds for the $k$-independence number of a graph which is is the maximum size of a set of vertices at pairwise distance greater than $k$. We construct graphs that attain equality for our first bound and show that our second bound compares favorably to previous bounds on the $k$-independence number.

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arXiv2016

Independent sets in polarity graphs

Given a projective plane $Σ$ and a polarity $θ$ of $Σ$, the corresponding polarity graph is the graph whose vertices are the points of $Σ$, and two distinct points $p_1$ and $p_2$ are adjacent if $p_1$ is incident to $p_2^{ θ}$ in $Σ$. A well-known example of a polarity graph is the Erdős-Rényi orthogonal polarity graph $ER_q$, which appears frequently in a variety of extremal problems. Eigenvalue methods provide an upper bound on the independence number of any polarity graph. Mubayi and Williford showed that in the case of $ER_q$, the eigenvalue method gives the correct upper bound in order of magnitude. We prove that this is also true for other families of polarity graphs. This includes a family of polarity graphs for which the polarity is neither orthogonal nor unitary. We conjecture that any polarity graph of a projective plane of order $q$ has an independent set of size $Ω(q^{3/2})$. Some related results are also obtained.

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arXiv2015

Characterizing graphs of maximum principal ratio

The principal ratio of a connected graph, denoted $γ(G)$, is the ratio of the maximum and minimum entries of its first eigenvector. Cioabă and Gregory conjectured that the graph on $n$ vertices maximizing $γ(G)$ is a kite graph: a complete graph with a pendant path. In this paper we prove their conjecture.

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arXiv2015

On a problem of Neumann

A conjecture widely attributed to Neumann is that all finite non-desarguesian projective planes contain a Fano subplane. In this note, we show that any finite projective plane of even order which admits an orthogonal polarity contains a Fano subplane. The number of planes of order less than $n$ previously known to contain a Fano subplane was $O(\log n)$, whereas the number of planes of order less than $n$ that our theorem applies to is not bounded above by any polynomial in $n$.

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arXiv2015

On the distance spectra of graphs

The distance matrix of a graph $G$ is the matrix containing the pairwise distances between vertices. The distance eigenvalues of $G$ are the eigenvalues of its distance matrix and they form the distance spectrum of $G$. We determine the distance spectra of halved cubes, double odd graphs, and Doob graphs, completing the determination of distance spectra of distance regular graphs having exactly one positive distance eigenvalue. We characterize strongly regular graphs having more positive than negative distance eigenvalues. We give examples of graphs with few distinct distance eigenvalues but lacking regularity properties. We also determine the determinant and inertia of the distance matrices of lollipop and barbell graphs.

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arXiv2015

Increasing paths in edge-ordered graphs: the hypercube and random graphs

An edge-ordering of a graph $G=(V,E)$ is a bijection $ϕ:E\to\{1,2,...,|E|\}$. Given an edge-ordering, a sequence of edges $P=e_1,e_2,...,e_k$ is an increasing path if it is a path in $G$ which satisfies $ϕ(e_i)<ϕ(e_j)$ for all $i<j$. For a graph $G$, let $f(G)$ be the largest integer $\ell$ such that every edge-ordering of $G$ contains an increasing path of length $\ell$. The parameter $f(G)$ was first studied for $G=K_n$ and has subsequently been studied for other families of graphs. This paper gives bounds on $f$ for the hypercube and the random graph $G(n,p)$.

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arXiv2015

Small dense subgraphs of polarity graphs and the extremal number for the 4-cycle

In this note, we show that for any $m \in \{1,2, \dots , q +1 \}$, if $G$ is a polarity graph of a projective plane of order $q$ that has an oval, then $G$ contains a subgraph on $m + \binom{m}{2}$ vertices with $m^2+\frac{m^4}{8q} - O ( \frac{m^4}{q^{3/2} } +m )$ edges. As an application, we give the best known lower bounds on the Turán number $\mathrm{ex}(n, C_4)$ for certain values of $n$. In particular, we disprove a conjecture of Abreu, Balbuena, and Labbate concerning $\mathrm{ex}(q^2-q-2, C_4)$ where $q$ is a power of $2$.

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arXiv2014

Orthogonal polarity graphs and Sidon sets

Determining the maximum number of edges in an $n$-vertex $C_4$-free graph is a well-studied problem that dates back to a paper of Erdős from 1938. One of the most important families of $C_4$-free graphs are the Erdős-Rényi orthogonal polarity graphs. We show that the Cayley sum graph constructed using a Bose-Chowla Sidon set is isomorphic to a large induced subgraph of the Erdős-Rényi orthogonal polarity graph. Using this isomorphism we prove that the Petersen graph is a subgraph of every sufficiently large Erdős-Rényi orthogonal polarity graph.

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arXiv2014

On sets of integers with restrictions on their products

A {\em product-injective labeling} of a graph $G$ is an injection $χ: V(G) \to \mathbb{Z}$ such that $χ(u)χ(v) \not= χ(x)χ(y)$ for any distinct edges $uv, xy\in E(G)$. Let $P(G)$ be the smallest $N \geq 1$ such that there exists a product-injective labeling $χ: V(G) \rightarrow [N]$. Let $P(n,d)$ be the maximum possible value of $P(G)$ over $n$-vertex graphs $G$ of maximum degree at most $d$. In this paper, we determine the asymptotic value of $P(n,d)$ for all but a small range of values of $d$ relative to $n$. Specifically, we show that there exist constants $a,b > 0$ such that $P(n,d) \sim n$ if $d \leq \sqrt{n}(\log n)^{-a}$ and $P(n,d) \sim n\log n$ if $d \geq \sqrt{n}(\log n)^{b}$.

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arXiv2014

On Coupon Colorings of Graphs

Let $G$ be a graph with no isolated vertices. A {\em $k$-coupon coloring} of $G$ is an assignment of colors from $[k] := \{1,2,\dots,k\}$ to the vertices of $G$ such that the neighborhood of every vertex of $G$ contains vertices of all colors from $[k]$. The maximum $k$ for which a $k$-coupon coloring exists is called the {\em coupon coloring number} of $G$, and is denoted $χ_{c}(G)$. In this paper, we prove that every $d$-regular graph $G$ has $χ_{c}(G) \geq (1 - o(1))d/\log d$ as $d \rightarrow \infty$, and the proportion of $d$-regular graphs $G$ for which $χ_c(G) \leq (1 + o(1))d/\log d$ tends to $1$ as $|V(G)| \rightarrow \infty$.

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arXiv2014

Sidon Sets and graphs without 4-cycles

The problem of determining the maximum number of edges in an $n$-vertex graph that does not contain a 4-cycle has a rich history in extremal graph theory. Using Sidon sets constructed by Bose and Chowla, for each odd prime power $q$ we construct a graph with $q^2 - q - 2$ vertices that does not contain a 4-cycle and has at least $\frac{1}{2}q^3 - q^2 - O(q^{3/4})$ edges. This disproves a conjecture of Abreu, Balbuena, and Labbate concerning the Turán number $\mathrm{ex}(q^2 - q - 2, C_4)$.

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arXiv2013

Asymptotic Improvement of the Gilbert-Varshamov Bound on the Size of Permutation Codes

Given positive integers $n$ and $d$, let $M(n,d)$ denote the maximum size of a permutation code of length $n$ and minimum Hamming distance $d$. The Gilbert-Varshamov bound asserts that $M(n,d) \geq n!/V(n,d-1)$ where $V(n,d)$ is the volume of a Hamming sphere of radius $d$ in $§_n$. Recently, Gao, Yang, and Ge showed that this bound can be improved by a factor $Ω(\log n)$, when $d$ is fixed and $n \to \infty$. Herein, we consider the situation where the ratio $d/n$ is fixed and improve the Gilbert-Varshamov bound by a factor that is \emph{linear in $n$}. That is, we show that if $d/n < 0.5$, then $$ M(n,d)\geq cn\,\frac{n!}{V(n,d-1)} $$ where $c$ is a positive constant that depends only on $d/n$. To establish this result, we follow the method of Jiang and Vardy. Namely, we recast the problem of bounding $M(n,d)$ into a graph-theoretic framework and prove that the resulting graph is locally sparse.

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