arXiv subjects

Michael Young

Publications and source records attributed to Michael Young.

At least 37 records · Page 2Linked to original sources

arXiv2018

Anti-Ramsey Multiplicities

The Ramsey multiplicity constant of a graph $H$ is the minimum proportion of copies of $H$ in the complete graph which are monochromatic under an edge-coloring of $K_n$ as $n$ goes to infinity. Graphs for which this minimum is asymptotically achieved by taking a random coloring are called {\em common}, and common graphs have been studied extensively, leading to the Burr-Rosta conjecture and Sidorenko's conjecture. Erdős and Sós asked what the maximum number of rainbow triangles is in a $3$-coloring of the edge set of $K_n$, a rainbow version of the Ramsey multiplicity question. A graph $H$ is called $r$-anti-common if the maximum proportion of rainbow copies of $H$ in any $r$-coloring of $E(K_n)$ is asymptotically achieved by taking a random coloring. In this paper, we investigate anti-Ramsey multiplicity for several families of graphs. We determine classes of graphs which are either anti-common or not. Some of these classes follow the same behavior as the monochromatic case, but some of them do not. In particular the rainbow equivalent of Sidorenko's conjecture, that all bipartite graphs are anti-common, is false.

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arXiv2018

Lower Bounds for the Exponential Domination Number of $C_m \times C_n$

A vertex $v$ in a porous exponential dominating set assigns weight $\left(\tfrac{1}{2}\right)^{dist(v,u)}$ to vertex $u$. A porous exponential dominating set of a graph $G$ is a subset of $V(G)$ such that every vertex in $V(G)$ has been assigned a sum weight of at least 1. In this paper the porous exponential dominating number, denoted by $γ_e^*(G)$, for the graph $G = C_m \times C_n$ is discussed. Anderson et. al. proved that $\frac{mn}{15.875}\le γ_e^*(C_m \times C_n) \le \frac{mn}{13}$ and conjectured that $\frac{mn}{13}$ is also the asymptotic lower bound. We use a linear programing approach to sharpen the lower bound to $\frac{mn}{13.7619 + ε(m,n)}$.

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arXiv2018

A linear programming method for exponential domination

For a graph $G,$ the set $D \subseteq V(G)$ is a porous exponential dominating set if $1 \le \sum_{d \in D} \left( 2 \right)^{1-dist(d,v)}$ for every $v \in V(G),$ where $dist(d,v)$ denotes the length of the shortest $dv$ path. The porous exponential dominating number of $G,$ denoted $γ_e^*(G),$ is the minimum cardinality of a porous exponential dominating set. For any graph $G,$ a technique is derived to determine a lower bound for $γ_e^*(G).$ Specifically for a grid graph $H,$ linear programing is used to sharpen bound found through the lower bound technique. Lower and upper bounds are determined for the porous exponential domination number of the King Grid $\mathcal{K_n},$ the Slant Grid $\mathcal{S_n},$ and the $n$-dimensional hypercube $Q_n.$

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arXiv2017

On exponential domination of the consecutive circulant graph

For a graph $G,$ we consider $D \subset V(G)$ to be a porous exponential dominating set if $1\le \sum_{d \in D}$ $\left( \frac{1}{2} \right)^{\text{dist}(d,v) -1}$ for every $v \in V(G),$ where dist$(d,v)$ denotes the length of the smallest $dv$ path. Similarly, $D \subset V(G)$ is a non-porous exponential dominating set is $1\le \sum_{d \in D} \left( \frac{1}{2} \right)^{\overline{\text{dist}}(d,v) -1}$ for every $v \in V(G),$ where $\overline{\text{dist}}(d,v)$ represents the length of the shortest $dv$ path with no internal vertices in $D.$ The porous and non-porous exponential dominating number of $G,$ denoted $γ_e^*(G)$ and $γ_e(G),$ are the minimum cardinality of a porous and non-porous exponential dominating set, respectively. The consecutive circulant graph, $C_{n, [\ell]},$ is the set of $n$ vertices such that vertex $v$ is adjacent to $v \pm i \mod n$ for each $i \in [\ell].$ In this paper we show $γ_e(C_{n, [\ell]}) = γ_e^*(C_{n, [\ell]}) = \left\lceil \tfrac{n}{3\ell +1} \right\rceil.$

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arXiv2017

Turán numbers of vertex-disjoint cliques in $r$-partite graphs

For two graphs $G$ and $H$, the Turán number $ex(G,H)$ is the maximum number of edges in a subgraph of $G$ that contains no copy of $H$. Chen, Li, and Tu determined the Turán numbers $ex(K_{m,n},kK_2)$ for all $k\geq 1$ [7]. In this paper we will determine the Turán numbers $ex(K_{a_1,\ldots,a_r},kK_r)$ for all $r\geq 3$ and $k\geq 1$.

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arXiv2017

The inverse eigenvalue problem of a graph: Multiplicities and minors

The inverse eigenvalue problem of a given graph $G$ is to determine all possible spectra of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in $G$. Barrett et al. introduced the Strong Spectral Property (SSP) and the Strong Multiplicity Property (SMP) in [8]. In that paper it was shown that if a graph has a matrix with the SSP (or the SMP) then a supergraph has a matrix with the same spectrum (or ordered multiplicity list) augmented with simple eigenvalues if necessary, that is, subgraph monotonicity. In this paper we extend this to a form of minor monotonicity, with restrictions on where the new eigenvalues appear. These ideas are applied to solve the inverse eigenvalue problem for all graphs of order five, and to characterize forbidden minors of graphs having at most one multiple eigenvalue.

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arXiv2017

The relationship between $k$-forcing and $k$-power domination

Zero forcing and power domination are iterative processes on graphs where an initial set of vertices are observed, and additional vertices become observed based on some rules. In both cases, the goal is to eventually observe the entire graph using the fewest number of initial vertices. Chang et al. introduced $k$-power domination in [Generalized power domination in graphs, {\it Discrete Applied Math.} 160 (2012) 1691-1698] as a generalization of power domination and standard graph domination. Independently, Amos et al. defined $k$-forcing in [Upper bounds on the $k$-forcing number of a graph, {\it Discrete Applied Math.} 181 (2015) 1-10] to generalize zero forcing. In this paper, we combine the study of $k$-forcing and $k$-power domination, providing a new approach to analyze both processes. We give a relationship between the $k$-forcing and the $k$-power domination numbers of a graph that bounds one in terms of the other. We also obtain results using the contraction of subgraphs that allow the parallel computation of $k$-forcing and $k$-power dominating sets.

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arXiv2016

Polychromatic colorings of complete graphs with respect to 1-,2-factors and Hamiltonian cycles

If G is a graph and H is a set of subgraphs of G, then an edge-coloring of G is called H-polychromatic if every graph from H gets all colors present in G on its edges. The H-polychromatic number of G, denoted poly_H(G), is the largest number of colors in an H-polychromatic coloring. In this paper, poly_H(G) is determined exactly when G is a complete graph and H is the family of all 1-factors. In addition poly_H(G) is found up to an additive constant term when G is a complete graph and H is the family of all 2-factors, or the family of all Hamiltonian cycles.

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arXiv2016

Power propagation time and lower bounds for power domination number

We present a counterexample to a lower bound for the power domination number given in Liao, Power domination with bounded time constraints, J. Comb. Optim. 31 (2016)725-742. We also define the power propagation time, using the power domination propagation ideas in Liao and the (zero forcing) propagation time in Hogben et al, Propagation time for zero forcing on a graph, Discrete Appl. Math.160 (2012) 1994-2005.

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arXiv2016

Note on von Neumann and Rényi entropies of a Graph

We conjecture that all connected graphs of order $n$ have von Neumann entropy at least as great as the star $K_{1,n-1}$ and prove this for almost all graphs of order $n$. We show that connected graphs of order $n$ have Rényi 2-entropy at least as great as $K_{1,n-1}$ and for $α>1$, $K_n$ maximizes Rényi $α$-entropy over graphs of order $n$. We show that adding an edge to a graph can lower its von Neumann entropy.

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arXiv2016

Multi-part Nordhaus-Gaddum type problems for tree-width, Colin de Verdière type parameters, and Hadwiger number

A traditional Nordhaus-Gaddum problem for a graph parameter $β$ is to find a (tight) upper or lower bound on the sum or product of $β(G)$ and $β(\bar{G})$ (where $\bar{G}$ denotes the complement of $G$). An $r$-decomposition $G_1,\dots,G_r$ of the complete graph $K_n$ is a partition of the edges of $K_n$ among $r$ spanning subgraphs $G_1,\dots,G_r$. A traditional Nordhaus-Gaddum problem can be viewed as the special case for $r=2$ of a more general $r$-part sum or product Nordhaus-Gaddum type problem. We determine the values of the $r$-part sum and product upper bounds asymptotically as $n$ goes to infinity for the parameters tree-width and its variants largeur d'arborescence, path-width, and proper path-width. We also establish ranges for the lower bounds for these parameters, and ranges for the upper and lower bounds of the $r$-part Nordhaus-Gaddum type problems for the parameters Hadwiger number, the Colin de Verdière number $μ$ that is used to characterize planarity, and its variants $ν$ and $ξ$.

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arXiv2016

Anti-van der Waerden numbers of 3-term arithmetic progressions

The \emph{anti-van der Waerden number}, denoted by $aw([n],k)$, is the smallest $r$ such that every exact $r$-coloring of $[n]$ contains a rainbow $k$-term arithmetic progression. Butler et. al. showed that $\lceil \log_3 n \rceil + 2 \le aw([n],3) \le \lceil \log_2 n \rceil + 1$, and conjectured that there exists a constant $C$ such that $aw([n],3) \le \lceil \log_3 n \rceil + C$. In this paper, we show this conjecture is true by determining $aw([n],3)$ for all $n$. We prove that for $7\cdot 3^{m-2}+1 \leq n \leq 21 \cdot 3^{m-2}$, \[ aw([n],3)=\left\{\begin{array}{ll} m+2, & \mbox{if $n=3^m$}\\ m+3, & \mbox{otherwise}. \end{array}\right.\]

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arXiv2016

Rainbow Arithmetic Progressions in Finite Abelian Groups

For positive integers $n$ and $k$, the \emph{anti-van der Waerden number} of $\mathbb{Z}_n$, denoted by $aw(\mathbb{Z}_n,k)$, is the minimum number of colors needed to color the elements of the cyclic group of order $n$ and guarantee there is a rainbow arithmetic progression of length $k$. Butler et al. showed a reduction formula for $aw(\mathbb{Z}_{n},3) = 3$ in terms of the prime divisors of $n$. In this paper, we analagously define the anti-van der Waerden number of a finite abelian group $G$ and show $aw(G,3)$ is determined by the order of $G$ and the number of groups with even order in a direct sum isomorphic to $G$. The \emph{unitary anti-van der Waerden number} of a group is also defined and determined.

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arXiv2016

Rainbow arithmetic progressions

In this paper, we investigate the anti-Ramsey (more precisely, anti-van der Waerden) properties of arithmetic progressions. For positive integers $n$ and $k$, the expression $aw([n],k)$ denotes the smallest number of colors with which the integers $\{1,\ldots,n\}$ can be colored and still guarantee there is a rainbow arithmetic progression of length $k$. We establish that $aw([n],3)=Θ(\log n)$ and $aw([n],k)=n^{1-o(1)}$ for $k\geq 4$. For positive integers $n$ and $k$, the expression $aw(Z_n,k)$ denotes the smallest number of colors with which elements of the cyclic group of order $n$ can be colored and still guarantee there is a rainbow arithmetic progression of length $k$. In this setting, arithmetic progressions can "wrap around," and $aw(Z_n,3)$ behaves quite differently from $aw([n],3)$, depending on the divisibility of $n$. As shown in [Jungić et al., \textit{Combin. Probab. Comput.}, 2003], $aw(Z_{2^m},3) = 3$ for any positive integer $m$. We establish that $aw(Z_n,3)$ can be computed from knowledge of $aw(Z_p,3)$ for all of the prime factors $p$ of $n$. However, for $k\geq 4$, the behavior is similar to the previous case, that is, $aw(Z_n,k)=n^{1-o(1)}$.

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arXiv2015

Fractional Zero Forcing via Three-color Forcing Games

An $r$-fold analogue of the positive semidefinite zero forcing process that is carried out on the $r$-blowup of a graph is introduced and used to define the fractional positive semidefinite forcing number. Properties of the graph blowup when colored with a fractional positive semidefinite forcing set are examined and used to define a three-color forcing game that directly computes the fractional positive semidefinite forcing number of a graph. We develop a fractional parameter based on the standard zero forcing process and it is shown that this parameter is exactly the skew zero forcing number with a three-color approach. This approach and an algorithm are used to characterize graphs whose skew zero forcing number equals zero.

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arXiv2014

The Chromatic Polynomials of Signed Petersen Graphs

Zaslavsky proved in 2012 that, up to switching isomorphism, there are six different signed Petersen graphs and that they could be told apart by their chromatic polynomials, by showing that the latter give distinct results when evaluated at 3. He conjectured that the six different signed Petersen graphs also have distinct zero-free chromatic polynomials, and that both types of chromatic polynomials have distinct evaluations at \emph{any} positive integer. We developed and executed a computer program (running in {\tt SAGE}) that efficiently determines the number of proper $k$-colorings for a given signed graph; our computations for the signed Petersen graphs confirm Zaslavsky's conjecture. We also computed the chromatic polynomials of all signed complete graphs with up to five vertices.

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arXiv2014

Propagation time for zero forcing on a graph

Zero forcing (also called graph infection) on a simple, undirected graph $G$ is based on the color-change rule: If each vertex of $G$ is colored either white or black, and vertex $v$ is a black vertex with only one white neighbor $w$, then change the color of $w$ to black. A minimum zero forcing set is a set of black vertices of minimum cardinality that can color the entire graph black using the color change rule. The propagation time of a zero forcing set $B$ of graph $G$ is the minimum number of steps that it takes to force all the vertices of $G$ black, starting with the vertices in $B$ black and performing independent forces simultaneously. The minimum and maximum propagation times of a graph are taken over all minimum zero forcing sets of the graph. It is shown that a connected graph of order at least two has more than one minimum zero forcing set realizing minimum propagation time. Graphs $G$ having extreme minimum propagation times $|G| - 1$, $|G| - 2$, and $0$ are characterized, and results regarding graphs having minimum propagation time $1$ are established. It is shown that the diameter is an upper bound for maximum propagation time for a tree, but in general propagation time and diameter of a graph are not comparable.

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arXiv2014

Crossing numbers of complete tripartite and balanced complete multipartite graphs

The crossing number cr(G) of a graph G is the minimum number of crossings in a nondegenerate planar drawing of G. The rectilinear crossing number cr'(G) of G is the minimum number of crossings in a rectilinear nondegenerate planar drawing (with edges as straight line segments) of G. Zarankiewicz proved in 1952 that cr'(K_{n_1,n_2})\le Z(n_1,n_2):= n_1/2*(n_1-1)/2*n_2/2*(n_2-1)/2. We define an analogous bound A(n_1,n_2,n_3) for the complete tripartite graph K_{n_1,n_2,n_3}, and prove that cr'(K_{n_1,n_2,n_3})\le A({n_1,n_2,n_3}). We also show that for n large enough, 0.973 A(n,n,n) \le cr'(K_{n,n,n}) and 0.666 A(n,n,n)\le cr(K_{n,n,n}), with the tighter rectilinear lower bound established through the use of flag algebras. A complete multipartite graph is balanced if the partite sets all have the same cardinality. We study asymptotic behavior of the crossing number of the balanced complete r-partite graph. Richter and Thomassen proved in 1997 that the limit as n\to\infty of cr(K_{n,n}) over the maximum number of crossings in a drawing of K_{n,n} exists and is at most 1/4. We define z(r)=3(r^2-r)/8(r^2+r-3) and show that for a fixed r and the balanced complete r-partite graph, z(r) is an upper bound to the limit superior of the crossing number divided by the maximum number of crossings in a drawing.

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