arXiv subjects

Daniel W. Cranston

Publications and source records attributed to Daniel W. Cranston.

At least 55 records · Page 3Linked to original sources

arXiv2020

The Iterated Local Directed Transitivity Model for Social Networks

We introduce a new directed graph model for social networks, based on the transitivity of triads. In the Iterated Local Directed Transitivity (ILDT) model, new nodes are born over discrete time-steps, and inherit the link structure of their parent nodes. The ILDT model may be viewed as a directed analogue of the ILT model for undirected graphs introduced in \cite{ilt}. We investigate network science and graph theoretical properties of ILDT digraphs. We prove that the ILDT model exhibits a densification power law, so that the digraphs generated by the models densify over time. The number of directed triads are investigated, and counts are given of the number of directed 3-cycles and transitive $3$-cycles. A higher number of transitive 3-cycles are generated by the ILDT model, as found in real-world, on-line social networks. In many instances of the chosen initial digraph, the model eventually generates graphs with Hamiltonian directed cycles. We finish with a discussion of the eigenvalues of the adjacency matrices of ILDT directed graphs, and provide further directions.

cs.SI↗

arXiv2019

Degeneracy and Colorings of Squares of Planar Graphs without 4-Cycles

We prove several results on coloring squares of planar graphs without 4-cycles. First, we show that if $G$ is such a graph, then $G^2$ is $(Δ(G)+72)$-degenerate. This implies an upper bound of $Δ(G)+73$ on the chromatic number of $G^2$ as well as on several variants of the chromatic number such as the list-chromatic number, paint number, Alon--Tarsi number, and correspondence chromatic number. We also show that if $Δ(G)$ is sufficiently large, then the upper bounds on each of these parameters of $G^2$ can all be lowered to $Δ(G)+2$ (which is best possible). To complement these results, we show that 4-cycles are unique in having this property. Specifically, let $S$ be a finite list of positive integers, with $4\notin S$. For each constant $C$, we construct a planar graph $G_{S,C}$ with no cycle with length in $S$, but for which $χ(G_{S,C}^2) > Δ(G_{S,C})+C$.

math.CO↗

arXiv2019

A Note on Bootstrap Percolation Thresholds in Plane Tilings using Regular Polygons

In \emph{$k$-bootstrap percolation}, we fix $p\in (0,1)$, an integer $k$, and a plane graph $G$. Initially, we infect each face of $G$ independently with probability $p$. Infected faces remain infected forever, and if a healthy (uninfected) face has at least $k$ infected neighbors, then it becomes infected. For fixed $G$ and $p$, the \emph{percolation threshold} is the largest $k$ such that eventually all faces become infected, with probability at least $1/2$. For a large class of infinite graphs, we show that this threshold is independent of $p$. We consider bootstrap percolation in tilings of the plane by regular polygons. A \emph{vertex type} in such a tiling is the cyclic order of the faces that meet a common vertex. First, we determine the percolation threshold for each of the Archimedean lattices. More generally, let $\mathcal{T}$ denote the set of plane tilings $T$ by regular polygons such that if $T$ contains one instance of a vertex type, then $T$ contains infinitely many instances of that type. We show that no tiling in $\mathcal{T}$ has threshold 4 or more. Further, the only tilings in $\mathcal{T}$ with threshold 3 are four of the Archimedean lattices. Finally, we describe a large subclass of $\mathcal{T}$ with threshold 2.

math.CO↗

arXiv2019

Coloring a graph with $Δ-1$ colors: Conjectures equivalent to the Borodin-Kostochka conjecture that appear weaker

Borodin and Kostochka conjectured that every graph $G$ with maximum degree $Δ\ge 9$ satisfies $χ\le \max\{ω, Δ-1\}$. We carry out an in-depth study of minimum counterexamples to the Borodin-Kostochka conjecture. Our main tool is the identification of graph joins that are $f$-choosable, where $f(v) = d(v) - 1$ for each vertex $v$. Since such a join cannot be an induced subgraph of a vertex critical graph with $χ= Δ$, we have a wealth of structural information about minimum counterexamples to the Borodin-Kostochka conjecture. Our main result proves that certain conjectures that are prima facie weaker than the Borodin-Kostochka conjecture are in fact equivalent to it. One such equivalent conjecture is the following: Any graph with $χ\ge Δ= 9$ contains $K_3 * \bar{K_6}$ as a subgraph.

math.CO↗

arXiv2019

The Hilton--Zhao Conjecture is True for Graphs with Maximum Degree 4

A simple graph $G$ is \emph{overfull} if $|E(G)|>Δ\lfloor|V(G)|/2\rfloor$. By the pigeonhole principle, every overfull graph $G$ has $χ'(G)>Δ$. The \emph{core} of a graph, denoted $G_Δ$, is the subgraph induced by its vertices of degree $Δ$. Vizing's Adjacency Lemma implies that if $χ'(G)>Δ$, then $G_Δ$ contains cycles. Hilton and Zhao conjectured that if $G_Δ$ has maximum degree 2 and $Δ\ge 4$, then $χ'(G)>Δ$ precisely when $G$ is overfull. We prove this conjecture for the case $Δ=4$.

math.CO↗

arXiv2019

A Characterization of $(4,2)$-Choosable Graphs

A graph $G$ is \emph{$(a,b)$-choosable} if given any list assignment $L$ with $|L(v)|=a$ for each $v\in V(G)$ there exists a function $φ$ such that $φ(v)\in L(v)$ and $|φ(v)|=b$ for all $v\in V(G)$, and whenever vertices $x$ and $y$ are adjacent $φ(x)\cap φ(y)=\emptyset$. Meng, Puleo, and Zhu conjectured a characterization of (4,2)-choosable graphs. We prove their conjecture.

math.CO↗

arXiv2019

Acyclic edge-coloring of planar graphs: $Δ$ colors suffice when $Δ$ is large

An \emph{acyclic edge-coloring} of a graph $G$ is a proper edge-coloring of $G$ such that the subgraph induced by any two color classes is acyclic. The \emph{acyclic chromatic index}, $χ'_a(G)$, is the smallest number of colors allowing an acyclic edge-coloring of $G$. Clearly $χ'_a(G)\ge Δ(G)$ for every graph $G$. Cohen, Havet, and Müller conjectured that there exists a constant $M$ such that every planar graph with $Δ(G)\ge M$ has $χ'_a(G)=Δ(G)$. We prove this conjecture.

math.CO↗

arXiv2018

Circular Flows in Planar Graphs

For integers $a\ge 2b>0$, a \emph{circular $a/b$-flow} is a flow that takes values from $\{\pm b, \pm(b+1), \dots, \pm(a-b)\}$. The Planar Circular Flow Conjecture states that every $2k$-edge-connected planar graph admits a circular $(2+\frac{2}{k})$-flow. The cases $k=1$ and $k=2$ are equivalent to the Four Color Theorem and Grötzsch's 3-Color Theorem. For $k\ge 3$, the conjecture remains open. Here we make progress when $k=4$ and $k=6$. We prove that (i) {\em every 10-edge-connected planar graph admits a circular 5/2-flow} and (ii) {\em every 16-edge-connected planar graph admits a circular 7/3-flow.} The dual version of statement (i) on circular coloring was previously proved by Dvořák and Postle (Combinatorica 2017), but our proof has the advantages of being much shorter and avoiding the use of computers for case-checking. Further, it has new implications for antisymmetric flows. Statement (ii) is especially interesting because the counterexamples to Jaeger's original Circular Flow Conjecture are 12-edge-connected nonplanar graphs that admit no circular 7/3-flow. Thus, the planarity hypothesis of (ii) is essential.

math.CO↗

arXiv2018

Proper Distinguishing Colorings with Few Colors for Graphs with Girth at Least 5

The distinguishing chromatic number, $χ_D(G)$, of a graph $G$ is the smallest number of colors in a proper coloring, $φ$, of $G$, such that the only automorphism of $G$ that preserves all colors of $φ$ is the identity map. Collins and Trenk conjectured that if $G$ is connected with girth at least 5 and $G\ne C_6$, then $χ_D(G)\le Δ+1$. We prove this conjecture.

math.CO↗

arXiv2018

Planar Graphs of Girth at least Five are Square $(Δ+ 2)$-Choosable

We prove a conjecture of Dvořák, Král, Nejedlý, and Škrekovski that planar graphs of girth at least five are square $(Δ+2)$-colorable for large enough $Δ$. In fact, we prove the stronger statement that such graphs are square $(Δ+2)$-choosable and even square $(Δ+2)$-paintable.

math.CO↗

arXiv2018

Subcubic edge chromatic critical graphs have many edges

We consider graphs $G$ with $Δ=3$ such that $χ'(G)=4$ and $χ'(G-e)=3$ for every edge $e$, so-called \emph{critical} graphs. Jakobsen noted that the Petersen graph with a vertex deleted, $P^*$, is such a graph and has average degree only $\frac83$. He showed that every critical graph has average degree at least $\frac83$, and asked if $P^*$ is the only graph where equality holds. A result of Cariolaro and Cariolaro shows that this is true. We strengthen this average degree bound further. Our main result is that if $G$ is a subcubic critical graph other than $P^*$, then $G$ has average degree at least $\frac{46}{17}\approx2.706$. This bound is best possible, as shown by the Hajos join of two copies of $P^*$.

math.CO↗

arXiv2016

Short fans and the 5/6 bound for line graphs

In 2011, the second author conjectured that every line graph $G$ satisfies $χ(G)\le \max\{ω(G),\frac{5Δ(G)+8}{6}\}$. This conjecture is best possible, as shown by replacing each edge in a 5-cycle by $k$ parallel edges, and taking the line graph. In this paper we prove the conjecture. We also develop more general techniques and results that will likely be of independent interest, due to their use in attacking the Goldberg--Seymour conjecture.

math.CO↗

arXiv2016

An Introduction to the Discharging Method via Graph Coloring

We provide a "how-to" guide to the use and application of the Discharging Method. Our aim is not to exhaustively survey results proved by this technique, but rather to demystify the technique and facilitate its wider use, using applications in graph coloring as examples. Along the way, we present some new proofs and new problems.

math.CO↗

arXiv2016

Planar graphs are 9/2-colorable

We show that every planar graph $G$ has a 2-fold 9-coloring. In particular, this implies that $G$ has fractional chromatic number at most $\frac92$. This is the first proof (independent of the 4 Color Theorem) that there exists a constant $k<5$ such that every planar $G$ has fractional chromatic number at most $k$.

math.CO↗

arXiv2016

Planar Graphs have Independence Ratio at least 3/13

The 4 Color Theorem (4CT) implies that every $n$-vertex planar graph has an independent set of size at least $\frac{n}4$; this is best possible, as shown by the disjoint union of many copies of $K_4$. In 1968, Erdős asked whether this bound on independence number could be proved more easily than the full 4CT. In 1976 Albertson showed (independently of the 4CT) that every $n$-vertex planar graph has an independent set of size at least $\frac{2n}9$. Until now, this remained the best bound independent of the 4CT. Our main result improves this bound to $\frac{3n}{13}$.

math.CO↗

arXiv2016

Edge Lower Bounds for List Critical Graphs, via Discharging

A graph $G$ is $k$-critical if $G$ is not $(k-1)$-colorable, but every proper subgraph of $G$ is $(k-1)$-colorable. A graph $G$ is $k$-choosable if $G$ has an $L$-coloring from every list assignment $L$ with $|L(v)|=k$ for all $v$, and a graph $G$ is \emph{$k$-list-critical} if $G$ is not $(k-1)$-choosable, but every proper subgraph of $G$ is $(k-1)$-choosable. The problem of bounding (from below) the number of edges in a $k$-critical graph has been widely studied, starting with work of Gallai and culminating with the seminal results of Kostochka and Yancey, who essentially solved the problem. In this paper, we improve the best lower bound on the number of edges in a $k$-list-critical graph. Our proof uses the discharging method, which makes it simpler and more modular than previous work in this area.

math.CO↗

arXiv2015

Beyond Degree Choosability

Let $G$ be a connected graph with maximum degree $Δ$. Brooks' theorem states that $G$ has a $Δ$-coloring unless $G$ is a complete graph or an odd cycle. A graph $G$ is \emph{degree-choosable} if $G$ can be properly colored from its lists whenever each vertex $v$ gets a list of $d(v)$ colors. In the context of list coloring, Brooks' theorem can be strengthened to the following. Every connected graph $G$ is degree-choosable unless each block of $G$ is a complete graph or an odd cycle; such a graph $G$ is a \emph{Gallai tree}. This degree-choosability result was further strengthened to Alon--Tarsi orientations; these are orientations of $G$ in which the number of spanning Eulerian subgraphs with an even number of edges differs from the number with an odd number of edges. A graph $G$ is \emph{degree-AT} if $G$ has an Alon--Tarsi orientation in which each vertex has indegree at least 1. Alon and Tarsi showed that if $G$ is degree-AT, then $G$ is also degree-choosable. Hladky, Kral, and Schauz showed that a connected graph is degree-AT if and only if it is not a Gallai tree. In this paper, we consider pairs $(G,x)$ where $G$ is a connected graph and $x$ is some specified vertex in $V(G)$. We characterize pairs such that $G$ has no Alon--Tarsi orientation in which each vertex has indegree at least 1 and $x$ has indegree at least 2. When $G$ is 2-connected, the characterization is simple to state.

math.CO↗

arXiv2015

Modified Linear Programming and Class 0 Bounds for Graph Pebbling

Given a configuration of pebbles on the vertices of a connected graph $G$, a \emph{pebbling move} removes two pebbles from some vertex and places one pebble on an adjacent vertex. The \emph{pebbling number} of a graph $G$ is the smallest integer $k$ such that for each vertex $v$ and each configuration of $k$ pebbles on $G$ there is a sequence of pebbling moves that places at least one pebble on $v$. First, we improve on results of Hurlbert, who introduced a linear optimization technique for graph pebbling. In particular, we use a different set of weight functions, based on graphs more general than trees. We apply this new idea to some graphs from Hurlbert's paper to give improved bounds on their pebbling numbers. Second, we investigate the structure of Class 0 graphs with few edges. We show that every $n$-vertex Class 0 graph has at least $\frac53n - \frac{11}3$ edges. This disproves a conjecture of Blasiak et al. For diameter 2 graphs, we strengthen this lower bound to $2n - 5$, which is best possible. Further, we characterize the graphs where the bound holds with equality and extend the argument to obtain an identical bound for diameter 2 graphs with no cut-vertex.

math.CO↗