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Stephen G. Hartke

Publications and source records attributed to Stephen G. Hartke.

At least 19 recordsLinked to original sources
arXiv2026

List coloring $C_3$-free planar graphs with a sparse matching of restricted lists

A graph $G$ is $k$-choosable if it has a proper coloring for every $k$-list assignment. While every $C_3$-free planar graph is $4$-choosable, some of them are not $3$-choosable, as constructed by Voigt. Hu and Zhu conjectured that if $G$ is a $C_3$-free planar graph and $X \subseteq V(G)$ induces a bipartite subgraph, then $G$ has a proper $L$-coloring whenever $|L(x)| = 3$ for $x \in X$ and $|L(v)| = 4$ for $v \in V(G) \setminus X$. As evidence, they proved the conjecture when $X$ is an independent set. We provide further evidence by proving the conjecture when the induced subgraph $G[X]$ is an induced sparse matching. This is the first result supporting the conjecture in which the set $X$ receiving smaller lists may induce a subgraph with edges.

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arXiv2026

Small $q$-kernels in digraphs with minimum in-degree $δ$

For a digraph $D$, a subset $Q\subseteq V(D)$ is called a $q$-kernel if $Q$ is an independent set and all vertices in $V(D)$ are reachable from $Q$ via a directed path of length at most $q$. Given integers $q\geq 2$ and $δ\geq 1$, Spiro arXiv:2404.07305 [math.CO] posed the question: what is the smallest constant $c_{δ,q}$ such that every digraph $D$ with minimum in-degree $δ$ has a $q$-kernel of size at most $c_{δ,q}|V(D)|$? We show the constants $c_{δ,q}$ are monotone in both $δ$ and $q$, and we improve upon the known upper bounds for $c_{δ,q}$. Our main results show $\frac{1}{δ+1} \leq c_{δ,q}\leq \frac{1}{\lfloor\sqrt{δ+1}\rfloor+1}$ for all $q \geq 3$ and $δ\geq 1$, and $ c_{δ,q}=\frac{1}{δ+1}$ whenever $δ\geq 1$ and $q \geq \left\lceil\frac{3δ}{2}\right\rceil + 1$.

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arXiv2025

Reconstruction of C_4-free graphs from the set of closed neighborhoods and digital convexity

Fomin, Kratochvíl, Lokshtanov, Mancini, and Telle showed that every $C_{4}$-free graph is reconstructible from the \emph{multiset} of closed neighborhoods. We strengthen their result proving that every $C_{4}$-free graph is reconstructible from the \emph{set} of closed neighborhoods. This extends the work of Lafrance et al.\ by showing that all $C_{4}$-free graphs, and hence all graphs of girth at least five, are reconstructible from their digitally convex sets. A subset $S$ of vertices in a graph $G$ is digitally convex if, for every vertex $v \notin S$, there is a private neighbor of $v$. We establish that reconstruction from digitally convex sets is equivalent to reconstruction from the set of closed neighborhoods.

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arXiv2025

Trail Trap: a variant of Partizan Edge Geography

We study a two-player game played on undirected graphs called {\sc Trail Trap}, which is a variant of a game known as {\sc Partizan Edge Geography}. One player starts by choosing any edge and moving a token from one endpoint to the other; the other player then chooses a different edge and does the same. Alternating turns, each player moves their token along an unused edge from its current vertex to an adjacent vertex, until one player cannot move and loses. We present an algorithm to determine which player has a winning strategy when the graph is a tree and partially characterize the trees on which a given player wins. Additionally, we show that it is NP-hard to determine if Player~2 has a winning strategy on {\sc Trail Trap} from the starting position, even for connected bipartite planar graphs with maximum degree $4$. We determine which player has a winning strategy for certain subclasses of complete bipartite graphs and grid graphs, and we propose several open problems for further study.

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arXiv2024

The forb-flex method for odd coloring and proper conflict-free coloring of planar graphs

We introduce a new tool useful for greedy coloring, which we call the forb-flex method, and apply it to odd coloring and proper conflict-free coloring of planar graphs. The odd chromatic number, denoted $χ_{\mathsf{o}}(G)$, is the smallest number of colors needed to properly color $G$ such that every non-isolated vertex of $G$ has a color appearing an odd number of times in its neighborhood. The proper conflict-free chromatic number, denoted $χ_{\mathsf{PCF}}(G)$, is the smallest number of colors needed to properly color $G$ such that every non-isolated vertex of $G$ has a color appearing uniquely in its neighborhood. Our new tool works by carefully counting the structures in the neighborhood of a vertex and determining if a neighbor of a vertex can be recolored at the end of a greedy coloring process to avoid conflicts. Combining this with the discharging method allows us to prove $χ_{\mathsf{PCF}}(G) \leq 4$ for planar graphs of girth at least 11, and $χ_{\mathsf{o}}(G) \leq 4$ for planar graphs of girth at least 10. These results improve upon the recent works of Cho, Choi, Kwon, and Park.

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arXiv2023

Domination of subcubic planar graphs with large girth

Since Reed conjectured in 1996 that the domination number of a connected cubic graph of order $n$ is at most $\lceil \frac13 n \rceil$, the domination number of cubic graphs has been extensively studied. It is now known that the conjecture is false in general, but Henning and Dorbec showed that it holds for graphs with girth at least $9$. Zhu and Wu stated an analogous conjecture for 2-connected cubic planar graphs. In this paper, we present a new upper bound for the domination number of subcubic planar graphs: if $G$ is a subcubic planar graph with girth at least 8, then $γ(G) < n_0 + \frac{3}{4} n_1 + \frac{11}{20} n_2 + \frac{7}{20} n_3$, where $n_i$ denotes the number of vertices in $G$ of degree $i$, for $i \in \{0,1,2,3\}$. We also prove that if $G$ is a subcubic planar graph with girth at least 9, then $γ(G) < n_0 + \frac{13}{17} n_1 + \frac{9}{17} n_2 + \frac{6}{17} n_3$.

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arXiv2021

A Classification of Hyperfocused 12-Arcs

A $k$-arc in PG($2,q$) is a set of $k$ points no three of which are collinear. A hyperfocused $k$-arc is a $k$-arc in which the $k \choose 2$ secants meet some external line in exactly $k-1$ points. Hyperfocused $k$-arcs can be viewed as 1-factorizations of the complete graph $K_k$ that embed in PG($2,q$). We study the 526,915,620 1-factorizations of $K_{12}$, determine which are embeddable in PG($2,q$), and classify hyperfocused $12$-arcs. Specifically we show if a $12$-arc $\mathcal{K}$ is a hyperfocused arc in PG($2,q$) then $q = 2^{5k}$ and $\mathcal{K}$ is a subset of a hyperconic including the nucleus.

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arXiv2017

Uniquely $K^{(k)}_r$-saturated Hypergraphs

In this paper we generalize the concept of uniquely $K_r$-saturated graphs to hypergraphs. Let $K_r^{(k)}$ denote the complete $k$-uniform hypergraph on $r$ vertices. For integers $k,r,n$ such that $2\le k r$. This is in contrast to the case $k=2$ and $r=3$ where only the Moore graphs of diameter two have this property. Our other construction keeps $n-r$ fixed; in this case we show that for any fixed $k\ge 2$ there can only be finitely many examples. We give a range for $n$ where these hypergraphs exist. For $n-r=1$ the range is completely determined: $k+1\le n \le {(k+2)^2\over 4}$. For larger values of $n-r$ the upper end of our range reaches approximately half of its upper bound. The lower end depends on the chromatic number of certain Johnson graphs.

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arXiv2017

Navigating Between Packings of Graphic Sequences

Let $π_1=(d_1^{(1)}, \ldots,d_n^{(1)})$ and $π_2=(d_1^{(2)},\ldots,d_n^{(2)})$ be graphic sequences. We say they \emph{pack} if there exist edge-disjoint realizations $G_1$ and $G_2$ of $π_1$ and $π_2$, respectively, on vertex set $\{v_1,\dots,v_n\}$ such that for $j\in\{1,2\}$, $d_{G_j}(v_i)=d_i^{(j)}$ for all $i\in\{1,\ldots,n\}$. In this case, we say that $(G_1,G_2)$ is a $(π_1,π_2)$-\textit{packing}. A clear necessary condition for graphic sequences $π_1$ and $π_2$ to pack is that $π_1+π_2$, their componentwise sum, is also graphic. It is known, however, that this condition is not sufficient, and furthermore that the general problem of determining if two sequences pack is $NP$- complete. S.~Kundu proved in 1973 that if $π_2$ is almost regular, that is each element is from $\{k-1, k\}$, then $π_1$ and $π_2$ pack if and only if $π_1+π_2$ is graphic. In this paper we will consider graphic sequences $π$ with the property that $π+\mathbf{1}$ is graphic. By Kundu's theorem, the sequences $π$ and $\mathbf{1}$ pack, and there exist edge-disjoint realizations $G$ and $\mathcal{I}$, where $\mathcal{I}$ is a 1-factor. We call such a $(π,\mathbf{1})$ packing a {\em Kundu realization}. Assume that $π$ is a graphic sequence, in which each term is at most $n/24$, that packs with $\mathbf{1}$. This paper contains two results. On one hand, any two Kundu realizations of the degree sequence $π+\mathbf{1}$ can be transformed into each other through a sequence of other Kundu realizations by swap operations. On the other hand, the same conditions ensure that any particular 1-factor can be part of a Kundu realization of $π+\mathbf{1}$.

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arXiv2017

Graph realizations constrained by skeleton graphs

In 2008 Amanatidis, Green and Mihail introduced the Joint Degree Matrix (JDM) model to capture the fundamental difference in assortativity of networks in nature studied by the physical and life sciences and social networks studied in the social sciences. In 2014 Czabarka proposed a direct generalization of the JDM model, the Partition Adjacency Matrix (PAM) model. In the PAM model the vertices have specified degrees, and the vertex set itself is partitioned into classes. For each pair of vertex classes the number of edges between the classes in a graph realization is prescribed. In this paper we apply the new {\em skeleton graph} model to describe the same information as the PAM model. Our model is more convenient for handling problems with low number of partition classes or with special topological restrictions among the classes. We investigate two particular cases in detail: (i) when there are only two vertex classes and (ii) when the skeleton graph contains at most one cycle.

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arXiv2015

On the Strong Chromatic Index of Sparse Graphs

The strong chromatic index of a graph $G$, denoted $χ_s'(G)$, is the least number of colors needed to edge-color $G$ so that edges at distance at most two receive distinct colors. The strong list chromatic index, denoted $χ_{s,\ell}'(G)$, is the least integer $k$ such that if arbitrary lists of size $k$ are assigned to each edge then $G$ can be edge-colored from those lists where edges at distance at most two receive distinct colors. We use the discharging method, the Combinatorial Nullstellensatz, and computation to show that if $G$ is a subcubic planar graph with $\operatorname{girth}(G) \geq 41$ then $χ_{s,\ell}'(G) \leq 5$, answering a question of Borodin and Ivanova [Precise upper bound for the strong edge chromatic number of sparse planar graphs, Discuss. Math. Graph Theory, 33(4), (2014) 759--770]. We further show that if $G$ is a subcubic planar graph and $\operatorname{girth}(G) \geq 30$, then $χ_s'(G) \leq 5$, improving a bound from the same paper. Finally, if $G$ is a planar graph with maximum degree at most four and $\operatorname{girth}(G) \geq 28$, then $χ_s'(G) \leq 7$, improving a more general bound of Wang and Zhao from [Odd graphs and its application on the strong edge coloring, arXiv:1412.8358] in this case.

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arXiv2014

Extending Precolorings to Distinguish Group Actions

Given a group $Γ$ acting on a set $X$, a $k$-coloring $ϕ:X\to\{1,\dots,k\}$ of $X$ is distinguishing with respect to $Γ$ if the only $γ\in Γ$ that fixes $ϕ$ is the identity action. The distinguishing number of the action $Γ$, denoted $D_Γ(X)$, is then the smallest positive integer $k$ such that there is a distinguishing $k$-coloring of $X$ with respect to $Γ$. This notion has been studied in a number of settings, but by far the largest body of work has been concerned with finding the distinguishing number of the action of the automorphism group of a graph $G$ upon its vertex set, which is referred to as the distinguishing number of $G$. The distinguishing number of a group action is a measure of how difficult it is to "break" all of the permutations arising from that action. In this paper, we aim to further differentiate the resilience of group actions with the same distinguishing number. In particular, we introduce a precoloring extension framework to address this issue. A set $S \subseteq X$ is a fixing set for $Γ$ if for every non-identity element $γ\in Γ$ there is an element $s \in S$ such that $γ(s) \neq s$. The distinguishing extension number $\operatorname{ext}_D(X,Γ;k)$ is the minimum number $m$ such that for all fixing sets $W \subseteq X$ with $|W| \geq m$, every $k$-coloring $c : X \setminus W \to [k]$ can be extended to a $k$-coloring that distinguishes $X$. In this paper, we prove that $\operatorname{ext}_D(\mathbb{R},\operatorname{Aut}(\mathbb{R}),2) =4$, where $\operatorname{Aut}(\mathbb{R})$ is comprised of compositions of translations and reflections. We also consider the distinguishing extension number of the circle and (finite) cycles, obtaining several exact results and bounds.

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arXiv2014

The Alpha Problem & Line Count Configurations

Motivated by the work of Chudnovsky and the Eisenbud-Mazur Conjecture on evolutions, Harbourne and Huneke give a series of conjectures that relate symbolic and regular powers of ideals of fat points in $\mathbb P^n$. The conjectures involve both containment statements and bounds for the initial degree in which there is a non-zero form in an ideal. Working with initial degrees, we verify two of these conjectures for special line count configurations in projective 2-space over an algebraically closed field of characteristic 0.

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arXiv2014

On the independence ratio of distance graphs

A distance graph is an undirected graph on the integers where two integers are adjacent if their difference is in a prescribed distance set. The independence ratio of a distance graph $G$ is the maximum density of an independent set in $G$. Lih, Liu, and Zhu [Star extremal circulant graphs, SIAM J. Discrete Math. 12 (1999) 491--499] showed that the independence ratio is equal to the inverse of the fractional chromatic number, thus relating the concept to the well studied question of finding the chromatic number of distance graphs. We prove that the independence ratio of a distance graph is achieved by a periodic set, and we present a framework for discharging arguments to demonstrate upper bounds on the independence ratio. With these tools, we determine the exact independence ratio for several infinite families of distance sets of size three, determine asymptotic values for others, and present several conjectures.

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arXiv2013

Minimal forbidden sets for degree sequence characterizations

Given a set $\mathcal{F}$ of graphs, a graph $G$ is $\mathcal{F}$-free if $G$ does not contain any member of $\mathcal{F}$ as an induced subgraph. Barrus, Kumbhat, and Hartke [M. D. Barrus, M. Kumbhat, and S. G. Hartke, Graph classes characterized both by forbidden subgraphs and degree sequences, J. Graph Theory (2008), no. 2, 131--148] called $\mathcal{F}$ a degree-sequence-forcing (DSF) set if, for each graph $G$ in the class $\mathcal{C}$ of $\mathcal{F}$-free graphs, every realization of the degree sequence of $G$ is also in $\mathcal{C}$. A DSF set is minimal if no proper subset is also DSF. In this paper, we present new properties of minimal DSF sets, including that every graph is in a minimal DSF set and that there are only finitely many DSF sets of cardinality $k$. Using these properties and a computer search, we characterize the minimal DSF triples.

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arXiv2013

A Branch-and-Cut Strategy for the Manickam-Miklos-Singhi Conjecture

The Manickam-Miklos-Singhi Conjecture states that when n is at least 4k, every multiset of n real numbers with nonnegative total sum has at least (n-1 choose k-1) k-subsets with nonnegative sum. We develop a branch-and-cut strategy using a linear programming formulation to show that verifying the conjecture for fixed values of k is a finite problem. To improve our search, we develop a zero-error randomized propagation algorithm. Using implementations of these algorithms, we verify a stronger form of the conjecture for all k at most seven.

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