arXiv subjects

Owen Henderschedt

Publications and source records attributed to Owen Henderschedt.

10 recordsLinked to original sources

arXiv2026

A finite victory over de Bruijn-Erdős in interval discrepancy

We study a finite form of the classical interval discrepancy problem. Starting from the unit interval, one repeatedly splits an existing interval into two until $n$ intervals have been produced. The discrepancy of such a process is the maximum, over all intermediate stages, of the ratio between the longest interval and the shortest interval. A theorem of de Bruijn and Erdős from 1949 shows that this ratio must approach $2$ as $n\to\infty$, and they give a sharp construction achieving this bound. For fixed $n$, their construction gives the upper bound $\operatorname{disc}(n)\leq 2-\frac{3}{2n}+O\bigl(\frac 1{n^2}\bigr)$. In this paper, we prove that $\operatorname{disc}(n)=2^{1-1/\lceil n/2\rceil}=2-\frac{4\ln 2}{n}+O\bigl(\frac 1{n^2}\bigr)$ for every $n$.

math.CO↗

arXiv2026

Yes, $(2K_2, K_4)$-free graphs are recolorable

We prove that every $(2K_2,K_4)$-free graph is recolorable. Equivalently, for every such graph $G$ and every $\ell\geq χ(G)+1$, the reconfiguration graph of proper $\ell$-colorings of $G$, in which two colorings are adjacent if they differ on exactly one vertex, is connected. This resolves the final remaining open case in the classification of recolorable $(F_1,F_2)$-free graphs when $F_1$ and $F_2$ have at most four vertices.

math.CO↗

arXiv2026

On the recolorability of $(2K_2, K_4)$-free graphs

Given a graph $G$ and an integer $\ell>χ(G)$, the reconfiguration graph of the $\ell$-colorings of $G$ has as its vertices as the proper $\ell$-colorings of $G$, with an edge between two colorings whenever they differ on exactly one vertex. We say that $G$ is \emph{recolorable} if this reconfiguration graph is connected for every $\ell>χ(G)$. Belavadi and Cameron determined which $(F_1,F_2)$-free graphs are recolorable whenever $F_1$ and $F_2$ are graphs on at most four vertices, with the single exception of $(2K_2,K_4)$-free graphs. Gaspers and Huang showed such graphs are $4$-colorable. The $3$-colorable case within this class has also been resolved, leaving the open question of whether every $(2K_2,K_4)$-free graph with chromatic number $4$ is recolorable. In this paper, we provide evidence toward an affirmative answer by establishing recolorability for three subclasses: $(2K_2,K_4,C_5)$-free graphs, $(2K_2,K_4,H_a,H_b)$-free graphs for any distinct $a,b\in \{2,3,4\}$, and $(2K_2,K_4,H_4)$-free graphs containing an induced $W_5$, where $H_i$ denotes the unique $2K_2$-free graph obtained from a $W_5$ by keeping exactly $i$ edges from the universal vertex to the cycle.

math.CO↗

arXiv2025

Shrinking the Jung radius: Maximizing partial coverage of finite point sets

Jung's theorem says that planar sets of diameter $1$ can be covered by a closed circular disk of radius $\frac 1{\sqrt3}$. In this paper we consider a fractional Jung-type problem for finite planar point-sets. Let $\mathcal{P}_n$ be the family of all finite sets of $n$ points in the plane, of diameter at most $1$. Let the function value $N_n(r)$ ($0 < r \leq 1$) be the largest integer $k$ so that for every point set $P \in \mathcal{P}_n$ there is a closed circular disk of radius $r$ which covers at least $k$ points of $P$. We focus on the radii $r=\frac 12$ and $r=\frac 14$ and prove exact maximum values. Concerning the radius $r= \frac 12$, we prove $N_n(\frac{1}{2})=\lceil \frac{n}{3}\rceil+1$. Concerning the radius $r= \frac 14$, we prove that $N_{n}(\frac{1}{4}) = \lceil \frac{n}{7}\rceil$ if $n$ is not a multiple of 7, and $N_{n}(\frac{1}{4})$ is $ \frac{n}{7}$ or $ \frac{n}{7}+1$ otherwise. We also initiate further study of the function $N_n(r)$ by giving lower and upper bounds for $N_n(r)$ ($0 < r < \frac 1{\sqrt3}$).

math.CO↗

arXiv2025

Extending total colorings in planar graphs

We initiate the study of total-coloring extensions, and focus our attention on planar graphs, asking: ``When can a total-$k$-coloring of some subgraph $H$ of a planar graph $G$ be extended to a total-$k$-coloring of $G$?'' We prove that if $H$ is a matching, then any total-$(Δ+3)$-coloring of $H$ in $G$ extends to $G$ provided $Δ\geq 28$; this number of colors is best-possible without introducing a distance condition on $H$. We also prove that if $H$ is a set of distance-3 cliques then any total-$(Δ+1)$-coloring of $H$ extends to $G$ provided $Δ\geq 27$; this distance condition cannot be lowered.

math.CO↗

arXiv2025

Odd Ramsey numbers of multipartite graphs and hypergraphs

Given a hypergraph $G$ and a subhypergraph $H$ of $G$, the \emph{odd Ramsey number} $r_{odd}(G,H)$ is the minimum number of colors needed to edge-color $G$ so that every copy of $H$ intersects some color class in an odd number of edges. Generalizing a result of \cite{BHZ} in two different ways, in this paper we prove $r_{odd} \left(K_{n,n}, K_{2,t} \right)=\frac{n}{t} + o(n)$ for all $t\geq 2$, and $r_{odd} \left(\mathcal{K}^{(k)}_{n,\dots,n}, \mathcal{K}_{1,\dots,1,2,2} \right) = \frac{n}{2} + o(n)$ for all $k\geq 2$. The latter is the first result studying odd Ramsey numbers for hypergraphs.

math.CO↗

arXiv2025

Total coloring graphs with large minimum degree

We prove that for all $\varepsilon>0$, there exists a positive integer $n_0$ such that if $G$ is a graph on $n\geq n_0$ vertices with $δ(G)\geq\tfrac{1}{2}(1 + \varepsilon)n$, then $G$ satisfies the Total Coloring Conjecture, that is, $χ_T(G)\leq Δ(G)+2$.

math.CO↗

arXiv2025

Graphs generated from minimal sets of finite point-set topologies

In 1985, Golumbic and Scheinerman established an equivalence between comparability graphs and containment graphs, graphs whose vertices represent sets, with edges indicating set containment. A few years earlier, McMorris and Zaslavsky characterized upper bound graphs, those derived from partially ordered sets where two elements share an edge if they have a common upper bound, by a specific edge clique cover condition. In this paper, we introduce a unifying framework for these results using finite point set topologies. Given a finite topology, we define a graph whose vertices correspond to its elements, with edges determined by intersections of their minimal containing sets, where intersection is understood in terms of the topological separation axioms. This construction yields a natural sequence of graph classes, one for each separation axiom, that connects and extends both classical results in a structured and intuitive way.

math.CO↗

arXiv2024

On orientations with forbidden out-degrees

Let $G$ be a $d$-regular graph and let $F\subseteq\{0, 1, 2, \ldots, d\}$ be a list of forbidden out-degrees. Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati conjectured that if $|F|<\tfrac{1}{2}d$, then $G$ should admit an $F$-avoiding orientation, i.e., an orientation where no out-degrees are in the forbidden list $F$. The conjecture is known for $d\leq 4$ due to work of Ma and Lu, and here we extend this to $d\leq 6$. The conjecture has also been studied in a generalized version, where $d, F$ are changed from constant values to functions $d(v), F(v)$ that vary over all $v\in V(G)$. We provide support for this generalized version by verifying it for some new cases, including when $G$ is 2-degenerate and when every $F(v)$ has some specific structure.

math.CO↗

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.

math.CO↗