arXiv subjects
Publications and source records attributed to Danny Rorabaugh.
Let $G$ be a simple graph and $v$ be a vertex of $G$. The triangle-degree of $v$ in $G$ is the number of triangles that contain $v$. While every graph has at least two vertices with the same degree, there are graphs in which every vertex has a distinct triangle-degree. In this paper, we construct an infinite family of graphs with this property. We also study the vertex degrees and size of graphs with this property.
Graph $G$ is $F$-saturated if $G$ contains no copy of graph $F$ but any edge added to $G$ produces at least one copy of $F$. One common variant of saturation is to remove the former restriction: $G$ is $F$-semi-saturated if any edge added to $G$ produces at least one new copy of $F$. In this paper we take this idea one step further. Rather than just allowing edges of $G$ to be in a copy of $F$, we require it: $G$ is $F$-covered if every edge of $G$ is in a copy of $F$. It turns out that there is smooth interaction between coverage and semi-saturation, which opens for investigation a natural analogue to saturation numbers. Therefore we present preliminary cover-saturation theory and structural bounds for the cover-saturation numbers of graphs. We also establish asymptotic cover-saturation densities for cliques and paths, and upper and lower bounds (with small gaps) for cycles and stars.
The current availability of soil moisture data over large areas comes from satellite remote sensing technologies (i.e., radar-based systems), but these data have coarse resolution and often exhibit large spatial information gaps. Where data are too coarse or sparse for a given need (e.g., precision agriculture), one can leverage machine-learning techniques coupled with other sources of environmental information (e.g., topography) to generate gap-free information and at a finer spatial resolution (i.e., increased granularity). To this end, we develop a spatial inference engine consisting of modular stages for processing spatial environmental data, generating predictions with machine-learning techniques, and analyzing these predictions. We demonstrate the functionality of this approach and the effects of data processing choices via multiple prediction maps over a United States ecological region with a highly diverse soil moisture profile (i.e., the Middle Atlantic Coastal Plains). The relevance of our work derives from a pressing need to improve the spatial representation of soil moisture for applications in environmental sciences (e.g., ecological niche modeling, carbon monitoring systems, and other Earth system models) and precision agriculture (e.g., optimizing irrigation practices and other land management decisions).
The PRIMAD model with its six components (i.e., Platform, Research Objective, Implementation, Methods, Actors, and Data), provides an abstract taxonomy to represent computational experiments and enforce reproducibility by design. In this paper, we assess the model applicability to a set of Laser Interferometer Gravitational-Wave Observatory (LIGO) workflows from literature sources (i.e., published papers). Our work outlines potentials and limits of the model in terms of its abstraction levels and application process.
We extend results regarding a combinatorial model introduced by Black, Drellich, and Tymoczko (2017+) which generalizes the folding of the RNA molecule in biology. Consider a word on alphabet $\{A_1, \overline{A}_1, \ldots, A_m, \overline{A}_m\}$ in which $\overline{A}_i$ is called the complement of $A_i$. A word $w$ is foldable if can be wrapped around a rooted plane tree $T$, starting at the root and working counterclockwise such that one letter labels each half edge and the two letters labeling the same edge are complements. The tree $T$ is called $w$-valid. We define a bijection between edge-colored plane trees and words folded onto trees. This bijection is used to characterize and enumerate words for which there is only one valid tree. We follow up with a characterization of words for which there exist exactly two valid trees. In addition, we examine the set $\mathcal{R}(n,m)$ consisting of all integers $k$ for which there exists a word of length $2n$ with exactly $k$ valid trees. Black, Drellich, and Tymoczko showed that for the $n$th Catalan number $C_n$, $\{C_n,C_{n-1}\}\subset \mathcal{R}(n,1)$ but $k\not\in\mathcal{R}(n,1)$ for $C_{n-1}<k<C_n$. We describe a superset of $\mathcal{R}(n,1)$ in terms of the Catalan numbers by which we establish more missing intervals. We also prove $\mathcal{R}(n,1)$ contains all non-negative integer less than $n+1$.
Word $W$ is said to encounter word $V$ provided there is a homomorphism $ϕ$ mapping letters to nonempty words so that $ϕ(V)$ is a substring of $W$. For example, taking $ϕ$ such that $ϕ(h)=c$ and $ϕ(u)=ien$, we see that "science" encounters "huh" since $cienc=ϕ(huh)$. The density of $V$ in $W$, $δ(V,W)$, is the proportion of substrings of $W$ that are homomorphic images of $V$. So the density of "huh" in "science" is $2/{8 \choose 2}$. A word is doubled if every letter that appears in the word appears at least twice. The dichotomy: Let $V$ be a word over any alphabet, $Σ$ a finite alphabet with at least 2 letters, and $W_n \in Σ^n$ chosen uniformly at random. Word $V$ is doubled if and only if $\mathbb{E}(δ(V,W_n)) \rightarrow 0$ as $n \rightarrow \infty$. We further explore convergence for nondoubled words and concentration of the limit distribution for doubled words around its mean.
The arc graph $δ(G)$ of a digraph $G$ is the digraph with the set of arcs of $G$ as vertex-set, where the arcs of $δ(G)$ join consecutive arcs of $G$. In 1981, Poljak and Rödl characterised the chromatic number of $δ(G)$ in terms of the chromatic number of $G$ when $G$ is symmetric (i.e., undirected). In contrast, directed graphs with equal chromatic numbers can have arc graphs with distinct chromatic numbers. Even though the arc graph of a symmetric graph is not symmetric, we show that the chromatic number of the iterated arc graph $δ^k(G)$ still only depends on the chromatic number of $G$ when $G$ is symmetric.
We investigate a correspondence between the complexity hierarchy of constraint satisfaction problems and a hierarchy of logical compactness hypotheses for finite relational structures. It seems that the harder a constraint satisfaction problem is, the stronger the corresponding compactness hypothesis is. At the top level, the NP-complete constraint satisfaction problems correspond to compactness hypotheses that are equivalent to the ultrafilter axiom in all the cases we have investigated. At the bottom level, the simplest constraint satisfaction problems correspond to compactness hypotheses that are readily provable from the axioms of Zermelo and Fraenkel.
An $(r-1,1)$-coloring of an $r$-regular graph $G$ is an edge coloring such that each vertex is incident to $r-1$ edges of one color and $1$ edge of a different color. In this paper, we completely characterize all $4$-regular pseudographs (graphs that may contain parallel edges and loops) which do not have a $(3,1)$-coloring. An $\{r-1,1\}$-factor of an $r$-regular graph is a spanning subgraph in which each vertex has degree either $r-1$ or $1$. We prove various conditions that that must hold for any vertex-minimal $5$-regular pseudographs without $(4,1)$-colorings or without $\{4,1\}$-factors. Finally, for each $r\geq 6$ we construct graphs that are not $(r-1,1)$-colorable and, more generally, are not $(r-t,t)$-colorable for small $t$.
A graph $G$ is $k$-$weighted-list-antimagic$ if for any vertex weighting $\omega\colon V(G)\to\mathbb{R}$ and any list assignment $L\colon E(G)\to2^{\mathbb{R}}$ with $|L(e)|\geq |E(G)|+k$ there exists an edge labeling $f$ such that $f(e)\in L(e)$ for all $e\in E(G)$, labels of edges are pairwise distinct, and the sum of the labels on edges incident to a vertex plus the weight of that vertex is distinct from the sum at every other vertex. In this paper we prove that every graph on $n$ vertices having no $K_1$ or $K_2$ component is $\lfloor{\frac{4n}{3}}\rfloor$-weighted-list-antimagic. An oriented graph $G$ is $k$-$oriented-antimagic$ if there exists an injective edge labeling from $E(G)$ into $\{1,\dotsc,|E(G)|+k\}$ such that the sum of the labels on edges incident to and oriented toward a vertex minus the sum of the labels on edges incident to and oriented away from that vertex is distinct from the difference of sums at every other vertex. We prove that every graph on $n$ vertices with no $K_1$ component admits an orientation that is $\lfloor{\frac{2n}{3}}\rfloor$-oriented-antimagic.
Word $W$ is an instance of word $V$ provided there is a homomorphism $\phi$ mapping letters to nonempty words so that $\phi(V) = W$. For example, taking $\phi$ such that $\phi(c)=fr$, $\phi(o)=e$ and $\phi(l)=zer$, we see that "freezer" is an instance of "cool". Let $\mathbb{I}_n(V,[q])$ be the probability that a random length $n$ word on the alphabet $[q] = \{1,2,\cdots q\}$ is an instance of $V$. Having previously shown that $\lim_{n \rightarrow \infty} \mathbb{I}_n(V,[q])$ exists, we now calculate this limit for two Zimin words, $Z_2 = aba$ and $Z_3 = abacaba$.
Free words are elements of a free monoid, generated over an alphabet via the binary operation of concatenation. Casually speaking, a free word is a finite string of letters. Henceforth, we simply refer to them as words. Motivated by recent advances in the combinatorial limit theory of graphs-notably those involving flag algebras, graph homomorphisms, and graphons-we investigate the extremal and asymptotic theory of pattern containment and avoidance in words. Word V is a factor of word W provided V occurs as consecutive letters within W. W is an instance of V provided there exists a nonerasing monoid homomorphsism ϕ with ϕ(V) = W. For example, using the homomorphism ϕ defined by ϕ(P) = Ror, ϕ(h) = a, and ϕ(D) = baugh, we see that Rorabaugh is an instance of PhD. W avoids V if no factor of W is an instance of V. V is unavoidable provided, over any finite alphabet, there are only finitely many words that avoid V. Unavoidable words were classified by Bean, Ehrenfeucht, and McNulty (1979) and Zimin (1982). We briefly address the following Ramsey-theoretic question: For unavoidable word V and a fixed alphabet, what is the longest a word can be that avoids V? The density of V in W is the proportion of nonempty substrings of W that are instances of V. Since there are 45 substrings in Rorabaugh and 28 of them are instances of PhD, the density of PhD in Rorabaugh is 28/45. We establish a number of asymptotic results for word densities, including the expected density of a word in arbitrarily long, random words and the minimum density of an unavoidable word over arbitrarily long words. This is joint work with Joshua Cooper.
How long can a word be that avoids the unavoidable? Word $W$ encounters word $V$ provided there is a homomorphism $ϕ$ defined by mapping letters to nonempty words such that $ϕ(V)$ is a subword of $W$. Otherwise, $W$ is said to avoid $V$. If, on any arbitrary finite alphabet, there are finitely many words that avoid $V$, then we say $V$ is unavoidable. Zimin (1982) proved that every unavoidable word is encountered by some word $Z_n$, defined by: $Z_1 = x_1$ and $Z_{n+1} = Z_n x_{n+1} Z_n$. Here we explore bounds on how long words can be and still avoid the unavoidable Zimin words.
A planar point set is in convex position precisely when it has a convex polygonization, that is, a polygonization with maximum interior angle measure at most π. We can thus talk about the convexity of a set of points in terms of the minimum, taken over all polygonizations, of the maximum interior angle. The main result presented here is a nontrivial combinatorial upper bound of this min-max value in terms of the number of points in the set. Motivated by a particular construction, we also pose a natural conjecture for the best upper bound.