arXiv subjects

Gregory J. Puleo

Publications and source records attributed to Gregory J. Puleo.

At least 19 recordsLinked to original sources

arXiv2021

A lower bound on the saturation number, and graphs for which it is sharp

Let $H$ be a fixed graph. We say that a graph $G$ is $H$-saturated if it has no subgraph isomorphic to $H$, but the addition of any edge to $G$ results in an $H$-subgraph. The saturation number $\mathrm{sat}(H,n)$ is the minimum number of edges in an $H$-saturated graph on $n$ vertices. Kászonyi and Tuza, in 1986, gave a general upper bound on the saturation number of a graph $H$, but a nontrivial lower bound has remained elusive. In this paper we give a general lower bound on $\mathrm{sat}(H,n)$ and prove that it is asymptotically sharp (up to an additive constant) on a large class of graphs. This class includes all threshold graphs and many graphs for which the saturation number was previously determined exactly. Our work thus gives an asymptotic common generalization of several earlier results. The class also includes disjoint unions of cliques, allowing us to address an open problem of Faudree, Ferrara, Gould, and Jacobson.

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arXiv2021

Strong coloring 2-regular graphs: Cycle restrictions and partial colorings

Let $H$ be a graph with $Δ(H) \leq 2$, and let $G$ be obtained from $H$ by gluing in vertex-disjoint copies of $K_4$. We prove that if $H$ contains at most one odd cycle of length exceeding $3$, or if $H$ contains at most $3$ triangles, then $χ(G) \leq 4$. This proves the Strong Coloring Conjecture for such graphs $H$. For graphs $H$ with $Δ=2$ that are not covered by our theorem, we prove an approximation result towards the conjecture.

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arXiv2019

On the Triangle Clique Cover and $K_t$ Clique Cover Problems

An edge clique cover of a graph is a set of cliques that covers all edges of the graph. We generalize this concept to "$K_t$ clique cover", i.e. a set of cliques that covers all complete subgraphs on $t$ vertices of the graph, for every $t \geq 1$. In particular, we extend a classical result of Erdös, Goodman, and Pósa (1966) on the edge clique cover number ($t = 2$), also known as the intersection number, to the case $t = 3$. The upper bound is tight, with equality holding only for the Turán graph $T(n,3)$. We also extend an algorithm of Scheinerman and Trenk (1999) to solve a weighted version of the $K_t$ clique cover problem on a superclass of chordal graphs. We also prove that the $K_t$ clique cover problem is NP-hard.

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arXiv2019

Upper bounds for inverse domination in graphs

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

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arXiv2019

Some results on multithreshold graphs

Jamison and Sprague defined a graph $G$ to be a $k$-threshold graph with thresholds $θ_1 , \ldots, θ_k$ (strictly increasing) if one can assign real numbers $(r_v)_{v \in V(G)}$, called ranks, such that for every pair of vertices $v,w$, we have $vw \in E(G)$ if and only if the inequality $θ_i \leq r_v + r_w$ holds for an odd number of indices $i$. When $k=1$ or $k=2$, the precise choice of thresholds $θ_1, \ldots, θ_k$ does not matter, as a suitable transformation of the ranks transforms a representation with one choice of thresholds into a representation with any other choice of thresholds. Jamison asked whether this remained true for $k \geq 3$ or whether different thresholds define different classes of graphs for such $k$, offering \$50 for a solution of the problem. Letting $C_t$ for $t > 1$ denote the class of $3$-threshold graphs with thresholds $-1, 1, t$, we prove that there are infinitely many distinct classes $C_t$, answering Jamison's question. We also consider some other problems on multithreshold graphs, some of which remain open.

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arXiv2019

Packing and covering directed triangles

We prove that if a directed multigraph $D$ has at most $t$ pairwise arc-disjoint directed triangles, then there exists a set of less than $2t$ arcs in $D$ which meets all directed triangles in $D$, except in the trivial case $t=0$. This answers affirmatively a question of Tuza from 1990.

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arXiv2018

Motif and Hypergraph Correlation Clustering

Motivated by applications in social and biological network analysis, we introduce a new form of agnostic clustering termed~\emph{motif correlation clustering}, which aims to minimize the cost of clustering errors associated with both edges and higher-order network structures. The problem may be succinctly described as follows: Given a complete graph $G$, partition the vertices of the graph so that certain predetermined `important' subgraphs mostly lie within the same cluster, while `less relevant' subgraphs are allowed to lie across clusters. Our contributions are as follows: We first introduce several variants of motif correlation clustering and then show that these clustering problems are NP-hard. We then proceed to describe polynomial-time clustering algorithms that provide constant approximation guarantees for the problems at hand. Despite following the frequently used LP relaxation and rounding procedure, the algorithms involve a sophisticated and carefully designed neighborhood growing step that combines information about both edge and motif structures. We conclude with several examples illustrating the performance of the developed algorithms on synthetic and real networks.

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arXiv2018

$t$-cores for $(Δ+t)$-edge-colouring

We extend the edge-coloring notion of core (subgraph induced by the vertices of maximum degree) to $t$-core (subgraph induced by the vertices $v$ with $d(v)+μ(v)> Δ+t$), and find a sufficient condition for $(Δ+t)$-edge-coloring. In particular, we show that for any $t\geq 0$, if the $t$-core of $G$ has multiplicity at most $t+1$, with its edges of multiplicity $t+1$ inducing a multiforest, then $χ'(G) \leq Δ+t$. This extends previous work of Ore, Fournier, and Berge and Fournier. A stronger version of our result (which replaces the multiforest condition with a vertex-ordering condition) generalizes a theorem of Hoffman and Rodger about cores of $Δ$-edge-colourable simple graphs. In fact, our bounds hold not only for chromatic index, but for the \emph{fan number} of a graph, a parameter introduced by Scheide and Stiebitz as an upper bound on chromatic index. We are able to give an exact characterization of the graphs $H$ such that $\mathrm{Fan}(G) \leq Δ(G)+t$ whenever $G$ has $H$ as its $t$-core.

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arXiv2018

Veto Interval Graphs and Variations

We introduce a variation of interval graphs, called veto interval (VI) graphs. A VI graph is represented by a set of closed intervals, each containing a point called a veto mark. The edge $ab$ is in the graph if the intervals corresponding to the vertices $a$ and $b$ intersect, and neither contains the veto mark of the other. We find families of graphs which are VI graphs, and prove results towards characterizing the maximum chromatic number of a VI graph. We define and prove similar results about several related graph families, including unit VI graphs, midpoint unit VI (MUVI) graphs, and single and double approval graphs. We also highlight a relationship between approval graphs and a family of tolerance graphs.

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arXiv2018

List-edge-colouring planar graphs with precoloured edges

Let $G$ be a simple planar graph of maximum degree $Δ$, let $t$ be a positive integer, and let $L$ be an edge list assignment on $G$ with $|L(e)| \geq Δ+t$ for all $e \in E(G)$. We prove that if $H$ is a subgraph of $G$ that has been $L$-edge-coloured, then the edge-precolouring can be extended to an $L$-edge-colouring of $G$, provided that $H$ has maximum degree $d\leq t$ and either $d \leq t-4$ or $Δ$ is large enough ($Δ\geq 16+d$ suffices). If $d>t$, there are examples for any choice of $Δ$ where the extension is impossible.

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arXiv2018

Extension from Precoloured Sets of Edges

We consider precolouring extension problems for proper edge-colourings of graphs and multigraphs, in an attempt to prove stronger versions of Vizing's and Shannon's bounds on the chromatic index of (multi)graphs in terms of their maximum degree $Δ$. We are especially interested in the following question: when is it possible to extend a precoloured matching to a colouring of all edges of a (multi)graph? This question turns out to be related to the notorious List Colouring Conjecture and other classic notions of choosability.

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arXiv2018

Paired Threshold Graphs

Threshold graphs are recursive deterministic network models that have been proposed for describing certain economic and social interactions. One drawback of this graph family is that it has limited generative attachment rules. To mitigate this problem, we introduce a new class of graphs termed Paired Threshold (PT) graphs described through vertex weights that govern the existence of edges via two inequalities. One inequality imposes the constraint that the sum of weights of adjacent vertices has to exceed a specified threshold. The second inequality ensures that adjacent vertices have a weight difference upper bounded by another threshold. We provide a conceptually simple characterization and decomposition of PT graphs, analyze their forbidden induced subgraphs and present a method for performing vertex weight assignments on PT graphs that satisfy the defining constraints. Furthermore, we describe a polynomial-time algorithm for recognizing PT graphs. We conclude our exposition with an analysis of the intersection number, diameter and clustering coefficient of PT graphs.

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arXiv2018

Graphs with $α_1$ and $τ_1$ both large

Given a graph $G$, let $τ_1(G)$ denote the smallest size of a set of edges whose deletion makes $G$ triangle-free, and let $α_1(G)$ denote the largest size of an edge set containing at most one edge from each triangle of $G$. Erdős, Gallai, and Tuza introduced several problems with the unifying theme that $α_1(G)$ and $τ_1(G)$ cannot both be "very large"; the most well-known such problem is their conjecture that $α_1(G) + τ_1(G) \leq |V(G)|^2/4$, which was proved by Norin and Sun. We consider three other problems within this theme (two introduced by Erdős, Gallai, and Tuza, another by Norin and Sun), all of which request an upper bound either on $\min\{α_1(G), τ_1(G)\}$ or on $α_1(G) + kτ_1(G)$ for some constant $k$, and prove the existence of graphs for which these quantities are "large".

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arXiv2017

The list chromatic index of simple graphs whose odd cycles intersect in at most one edge

We study the class of simple graphs $\mathcal{G}^*$ for which every pair of distinct odd cycles intersect in at most one edge. We give a structural characterization of the graphs in $\mathcal{G}^*$ and prove that every $G \in \mathcal{G}^*$ satisfies the list-edge-coloring conjecture. When $Δ(G) \geq 4$, we in fact prove a stronger result about kernel-perfect orientations in $L(G)$ which implies that $G$ is $(mΔ(G):m)$-edge-choosable and $Δ(G)$-edge-paintable for every $m \geq 1$.

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arXiv2017

On (4,2)-Choosable Graphs

A graph $G$ is called $(a,b)$-choosable if for any list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $a$ permissible colours, there is a $b$-tuple $L$-colouring of $G$. An $(a,1)$-choosable graph is also called $a$-choosable. In the pioneering paper on list colouring of graphs by Erdős, Rubin and Taylor, $2$-choosable graphs are characterized. Confirming a special case of a conjecture of Erdős--Rubin--Taylor, Tuza and Voigt proved that $2$-choosable graphs are $(2m,m)$-choosable for any positive integer $m$. On the other hand, Voigt proved that if $m$ is an odd integer, then these are the only $(2m,m)$-choosable graphs; however, when $m$ is even, there are $(2m,m)$-choosable graphs that are not $2$-choosable. A graph is called $3$-choosable-critical if it is not $2$-choosable, but all its proper subgraphs are $2$-choosable. Voigt conjectured that for every positive integer $m$, all bipartite $3$-choosable-critical graphs are $(4m,2m)$-choosable. In this paper, we determine which $3$-choosable-critical graphs are $(4,2)$-choosable, refuting Voigt's conjecture in the process. Nevertheless, a weaker version of the conjecture is true: we prove that there is an even integer $k$ such that for any positive integer $m$, every bipartite $3$-choosable-critical graph is $(2km,km)$-choosable. Moving beyond $3$-choosable-critical graphs, we present an infinite family of non-$3$-choosable-critical graphs which have been shown by computer analysis to be $(4,2)$-choosable. This shows that the family of all $(4,2)$-choosable graphs has rich structure.

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arXiv2017

Online Sum-Paintability: Slow-Coloring of Trees

The slow-coloring game is played by Lister and Painter on a graph $G$. On each round, Lister marks a nonempty subset $M$ of the remaining vertices, scoring $|M|$ points. Painter then gives a color to a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. Painter's goal is to minimize the total score; Lister seeks to maximize it. The score that each player can guarantee doing no worse than is the sum-color cost of $G$, written $\mathring{\rm s}(G)$. We develop a linear-time algorithm to compute $\mathring{\rm s}(G)$ when $G$ is a tree, enabling us to characterize the $n$-vertex trees with the largest and smallest values. Our algorithm also computes on trees the interactive sum choice number, a parameter recently introduced by Bonamy and Meeks.

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arXiv2017

The Interactive Sum Choice Number of Trees

We study the interactive sum choice number, a game coloring parameter introduced by Bonamy and Meeks, and obtain a recursive formula for the interactive sum choice number of forests. This formula coincides with a formula for the slow coloring cost of forests, a parameter introduced by Mahoney, Puleo, and West, and shows that these parameters are equal on forests. This answers a question of Bonamy and Meeks.

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arXiv2017

Online Paintability: The Slow-Coloring Game

The slow-coloring game is played by Lister and Painter on a graph $G$. On each round, Lister marks a nonempty subset $M$ of the uncolored vertices, scoring $|M|$ points. Painter then gives a color to a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. Painter and Lister want to minimize and maximize the total score, respectively. The best score that each player can guarantee is the sum-color cost of $G$, written $\mathring{\mathrm{s}}(G)$. The game is an online variant of online sum list coloring. We proe $\frac{|V(G)|}{2α(G)} + \frac{1}{2} \leq \frac{\mathring{\mathrm{s}}(G)}{|V(G)|} \leq \max\left\{ \frac{|V(H)|}{α(H)} : H \subset G\right\}$, where $α(G)$ is the independence number, and we study when equality holds in the bounds. We compute $\mathring{\mathrm{s}}(G)$ for graphs with $α(G) = 2$. Among $n$-vertex graphs, we prove that $\mathring{\mathrm{s}}$ is minimized by the star and maximized by the path. We also obtain good bounds on $\mathring{\mathrm{s}}(K_{r,s})$.

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