arXiv subjects

Hajo Broersma

Publications and source records attributed to Hajo Broersma.

17 recordsLinked to original sources

arXiv2025

Path Extendable Tournaments

A digraph $D$ is called \emph{path extendable} if for every nonhamiltonian (directed) path $P$ in $D$, there exists another path $P^\prime$ with the same initial and terminal vertices as $P$, and $V(P^\prime) = V (P)\cup \{w\}$ for a vertex $w \in V(D)\setminus V(P)$. Hence, path extendability implies paths of continuous lengths between every vertex pair. In earlier works of C. Thomassen and K. Zhang, it was shown that the condition of small $i(T)$ or positive $π_2(T)$ implies paths of continuous lengths between every vertex pair in a tournament $T$, where $i(T)$ is the irregularity of $T$ and $π_2(T)$ denotes for the minimum number of paths of length $2$ from $u$ to $v$ among all vertex pairs $\{u,v\}$. Motivated by these results, we study sufficient conditions in terms of $i(T)$ and $π_2(T)$ that guarantee a tournament $T$ is path extendable. We prove that (1) a tournament $T$ is path extendable if $i(T)< 2π_2(T)-(|T|+8)/6$, and (2) a tournament $T$ is path extendable if $π_2(T) > (7|T|-10)/36$. As an application, we deduce that almost all random tournaments are path extendable.

math.CO↗

arXiv2024

Closures and heavy pairs for hamiltonicity

We say that a graph $G$ on $n$ vertices is $\{H,F\}$-$o$-heavy if every induced subgraph of $G$ isomorphic to $H$ or $F$ contains two nonadjacent vertices with degree sum at least $n$. Generalizing earlier sufficient forbidden subgraph conditions for hamiltonicity, in 2012, Li, Ryjáček, Wang and Zhang determined all connected graphs $R$ and $S$ of order at least 3 other than $P_3$ such that every 2-connected $\{R,S\}$-$o$-heavy graph is hamiltonian. In particular, they showed that, up to symmetry, $R$ must be a claw and $S\in\{P_4,P_5,C_3,Z_1,Z_2,B,N,W\}$. In 2008, Čada extended Ryjáček's closure concept for claw-free graphs by introducing what we call the $c$-closure for claw-$o$-heavy graphs. We apply it here to characterize the structure of the $c$-closure of 2-connected $\{R,S\}$-$o$-heavy graphs, where $R$ and $S$ are as above. Our main results extend or generalize several earlier results on hamiltonicity involving forbidden or $o$-heavy subgraphs.

math.CO↗

arXiv2024

Connected Turán numbers for Berge paths in hypergraphs

Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs. Denote by $\ex^{\mathrm{conn}}_r(n,\mathcal{F})$ the maximum number of hyperedges in an $n$-vertex connected $r$-uniform hypergraph which contains no member of $\mathcal{F}$ as a subhypergraph. Denote by $\mathcal{B}C_k$ the Berge cycle of length $k$, and by $\mathcal{B}P_k$ the Berge path of length $k$. Füredi, Kostochka and Luo, and independently Győri, Salia and Zamora determined $\ex^{\mathrm{conn}}_r(n,\mathcal{B}P_k)$ provided $k$ is large enough compared to $r$ and $n$ is sufficiently large. For the case $k\le r$, Kostochka and Luo obtained an upper bound for $\ex^{\mathrm{conn}}_r(n,\mathcal{B}P_k)$. In this paper, we continue investigating the case $k\le r$. We precisely determine $\ex^{\mathrm{conn}}_r(n,\mathcal{B}P_k)$ when $n$ is sufficiently large and $n$ is not a multiple of~$r$. For the case $k=r+1$, we determine $\ex^{\mathrm{conn}}_r(n,\mathcal{B}P_k)$ asymptotically.

math.CO↗

arXiv2024

Growth rates of the bipartite Erdős-Gyárfás function

Given two graphs $G, H$ and a positive integer $q$, an $(H,q)$-coloring of $G$ is an edge-coloring of $G$ such that every copy of $H$ in $G$ receives at least $q$ distinct colors. The bipartite Erdős-Gyárfás function $r(K_{n,n}, K_{s,t}, q)$ is defined to be the minimum number of colors needed for $K_{n,n}$ to have a $(K_{s,t}, q)$-coloring. For balanced complete bipartite graphs $K_{p,p}$, the function $r(K_{n,n}, K_{p,p}, q)$ was studied systematically in [Axenovich, Füredi and Mubayi, {\it J. Combin. Theory Ser. B} {\bf 79} (2000), 66--86]. In this paper, we study the asymptotic behavior of this function for complete bipartite graphs $K_{s,t}$ that are not necessarily balanced. Our main results deal with thresholds and lower and upper bounds for the growth rate of this function, in particular for (sub)linear and (sub)quadratic growth. We also obtain new lower bounds for the balanced bipartite case, and improve several results given by Axenovich, Füredi and Mubayi. Our proof techniques are based on an extension to bipartite graphs of the recently developed Color Energy Method by Pohoata and Sheffer and its refinements, and a generalization of an old result due to Corrádi.

math.CO↗

arXiv2024

A note on generalized crowns in linear r-graphs

An $r$-graph $H$ is a hypergraph consisting of a nonempty set of vertices $V$ and a collection of $r$-element subsets of $V$ we refer to as the edges of $H$. An $r$-graph $H$ is called linear if any two edges of $H$ intersect in at most one vertex. Let $F$ and $H$ be two linear $r$-graphs. If $H$ contains no copy of $F$, then $H$ is called $F$-free. The linear Turán number of $F$, denoted by $ex_r^{lin}(n,F)$, is the maximum number of edges in any $F$-free $n$-vertex linear $r$-graph. The crown $C_{13}$ (or $E_4$) is a linear 3-graph which is obtained from three pairwise disjoint edges by adding one edge that intersects all three of them in one vertex. In 2022, Gyárfás, Ruszinkó and Sárközy initiated the study of $ex_3^{lin}(n,F)$ for different choices of an acyclic 3-graph $F$. They determined the linear Turán numbers for all linear 3-graphs with at most 4 edges, except the crown. They established lower and upper bounds for $ex_3^{lin}(n,C_{13})$. In fact, their lower bound on $ex_3^{lin}(n,C_{13})$ is essentially tight, as was shown in a recent paper by Tang, Wu, Zhang and Zheng. In this paper, we generalize the notion of a crown to linear $r$-graphs for $r\ge 3$, and also generalize the above results to linear $r$-graphs.

math.CO↗

arXiv2023

Online Graph Coloring with Predictions

We introduce learning augmented algorithms to the online graph coloring problem. Although the simple greedy algorithm FirstFit is known to perform poorly in the worst case, we are able to establish a relationship between the structure of any input graph $G$ that is revealed online and the number of colors that FirstFit uses for $G$. Based on this relationship, we propose an online coloring algorithm FirstFitPredictions that extends FirstFit while making use of machine learned predictions. We show that FirstFitPredictions is both \emph{consistent} and \emph{smooth}. Moreover, we develop a novel framework for combining online algorithms at runtime specifically for the online graph coloring problem. Finally, we show how this framework can be used to robustify by combining it with any classical online coloring algorithm (that disregards the predictions).

cs.DS↗

arXiv2023

Turán numbers of general hypergraph star forests

Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs, and let $H$ be an $r$-uniform hypergraph. Then $H$ is called $\mathcal{F}$-free if it does not contain any member of $\mathcal{F}$ as a subhypergraph. The Turán number of $\mathcal{F}$, denoted by $ex_r(n,\mathcal{F})$, is the maximum number of hyperedges in an $\mathcal{F}$-free $n$-vertex $r$-uniform hypergraph. Our current results are motivated by earlier results on Turán numbers of star forests and hypergraph star forests. In particular, Lidický, Liu and Palmer [Electron. J. Combin. 20 (2013)] determined the Turán number $ex(n,F)$ of a star forest $F$ for sufficiently large $n$. Recently, Khormali and Palmer [European. J. Combin. 102 (2022) 103506] generalized the above result to three different well-studied hypergraph settings, but restricted to the case that all stars in the hypergraph star forests are identical. We further generalize these results to general hypergraph star forests.

math.CO↗

arXiv2023

Sharp bounds for Laplacian spectral moments of digraphs with a fixed dichromatic number

The $k$-th Laplacian spectral moment of a digraph $G$ is defined as $\sum_{i=1}^n λ_i^k$, where $λ_i$ are the eigenvalues of the Laplacian matrix of $G$ and $k$ is a nonnegative integer. For $k=2$, this invariant is better known as the Laplacian energy of $G$. We extend recently published results by characterizing the digraphs which attain the minimal and maximal Laplacian energy within classes of digraphs with a fixed dichromatic number. We also determine sharp bounds for the third Laplacian spectral moment within the special subclass which we define as join digraphs. We leave the full characterization of the extremal digraphs for $k\ge 3$ as an open problem.

math.CO↗

arXiv2022

Integer colorings with no rainbow 3-term arithmetic progression

In this paper, we study the rainbow Erdős-Rothschild problem with respect to 3-term arithmetic progressions. We obtain the asymptotic number of $r$-colorings of $[n]$ without rainbow 3-term arithmetic progressions, and we show that the typical colorings with this property are 2-colorings. We also prove that $[n]$ attains the maximum number of rainbow 3-term arithmetic progression-free $r$-colorings among all subsets of $[n]$. Moreover, the exact number of rainbow 3-term arithmetic progression-free $r$-colorings of $\mathbb{Z}_p$ is obtained, where $p$ is any prime and $\mathbb{Z}_p$ is the cyclic group of order $p$.

math.CO↗

arXiv2021

On a conjecture of Nikiforov involving a spectral radius condition for a graph to contain all trees

We partly confirm a Brualdi-Solheid-Turán type conjecture due to Nikiforov, which is a spectral radius analogue of the well-known Erdős-Sós Conjecture that any tree of order $t$ is contained in a graph of average degree greater than $t-2$. We confirm Nikiforov's Conjecture for all brooms and for a larger class of spiders. For our proofs we also obtain a new Turán type result which might turn out to be of independent interest.

math.CO↗

arXiv2021

Spectral radius conditions for the existence of all subtrees of diameter at most four

Let $μ(G)$ denote the spectral radius of a graph $G$. We partly confirm a conjecture due to Nikiforov, which is a spectral radius analogue of the well-known Erdős-Sós Conjecture that any tree of order $t$ is contained in a graph of average degree greater than $t-2$. Let $S_{n,k}=K_{k}\vee\overline{K_{n-k}}$, and let $S_{n,k}^{+}$ be the graph obtained from $S_{n,k}$ by adding a single edge joining two vertices of the independent set of $S_{n,k}$. In 2010, Nikiforov conjectured that for a given integer $k$, every graph $G$ of sufficiently large order $n$ with $μ(G)\geq μ(S_{n,k}^{+})$ contains all trees of order $2k+3$, unless $G=S_{n,k}^{+}$. We confirm this conjecture for trees with diameter at most four, with one exception. In fact, we prove the following stronger result for $k\geq 8$. If a graph $G$ with sufficiently large order $n$ satisfies $μ(G)\geq μ(S_{n,k})$ and $G\neq S_{n,k}$, then $G$ contains all trees of order $2k+3$ with diameter at most four, except for the tree obtained from a star $K_{1,k+1}$ by subdividing each of its $k+1$ edges once.

math.CO↗

arXiv2021

The Erdős-Gyárfás function with respect to Gallai-colorings

For fixed $p$ and $q$, an edge-coloring of the complete graph $K_n$ is said to be a $(p, q)$-coloring if every $K_p$ receives at least $q$ distinct colors. The function $f(n, p, q)$ is the minimum number of colors needed for $K_n$ to have a $(p, q)$-coloring. This function was introduced about 45 years ago, but was studied systematically by Erdős and Gyárfás in 1997, and is now known as the Erdős-Gyárfás function. In this paper, we study $f(n, p, q)$ with respect to Gallai-colorings, where a Gallai-coloring is an edge-coloring of $K_n$ without rainbow triangles. Combining the two concepts, we consider the function $g(n, p, q)$ that is the minimum number of colors needed for a Gallai-$(p, q)$-coloring of $K_n$. Using the anti-Ramsey number for $K_3$, we have that $g(n, p, q)$ is nontrivial only for $2\leq q\leq p-1$. We give a general lower bound for this function and we study how this function falls off from being equal to $n-1$ when $q=p-1$ and $p\geq 4$ to being $Θ(\log n)$ when $q = 2$. In particular, for appropriate $p$ and $n$, we prove that $g=n-c$ when $q=p-c$ and $c\in \{1,2\}$, $g$ is at most a fractional power of $n$ when $q=\lfloor\sqrt{p-1}\rfloor$, and $g$ is logarithmic in $n$ when $2\leq q\leq \lfloor\log_2 (p-1)\rfloor+1$.

math.CO↗

arXiv2021

Dopant Network Processing Units: Towards Efficient Neural-network Emulators with High-capacity Nanoelectronic Nodes

The rapidly growing computational demands of deep neural networks require novel hardware designs. Recently, tunable nanoelectronic devices were developed based on hopping electrons through a network of dopant atoms in silicon. These "Dopant Network Processing Units" (DNPUs) are highly energy-efficient and have potentially very high throughput. By adapting the control voltages applied to its terminals, a single DNPU can solve a variety of linearly non-separable classification problems. However, using a single device has limitations due to the implicit single-node architecture. This paper presents a promising novel approach to neural information processing by introducing DNPUs as high-capacity neurons and moving from a single to a multi-neuron framework. By implementing and testing a small multi-DNPU classifier in hardware, we show that feed-forward DNPU networks improve the performance of a single DNPU from 77% to 94% test accuracy on a binary classification task with concentric classes on a plane. Furthermore, motivated by the integration of DNPUs with memristor arrays, we study the potential of using DNPUs in combination with linear layers. We show by simulation that a single-layer MNIST classifier with only 10 DNPUs achieves over 96% test accuracy. Our results pave the road towards hardware neural-network emulators that offer atomic-scale information processing with low latency and energy consumption.

cs.LG↗

arXiv2021

Extremal problems and results related to Gallai-colorings

A Gallai-coloring (Gallai-$k$-coloring) is an edge-coloring (with colors from $\{1, 2, \ldots, k\}$) of a complete graph without rainbow triangles. Given a graph $H$ and a positive integer $k$, the $k$-colored Gallai-Ramsey number $GR_k(H)$ is the minimum integer $n$ such that every Gallai-$k$-coloring of the complete graph $K_n$ contains a monochromatic copy of $H$. In this paper, we consider two extremal problems related to Gallai-$k$-colorings. First, we determine upper and lower bounds for the maximum number of edges that are not contained in any rainbow triangle or monochromatic triangle in a $k$-edge-coloring of $K_n$. Second, for $n\geq GR_k(K_3)$, we determine upper and lower bounds for the minimum number of monochromatic triangles in a Gallai-$k$-coloring of $K_{n}$, yielding the exact value for $k=3$. Furthermore, we determine the Gallai-Ramsey number $GR_k(K_4+e)$ for the graph on five vertices consisting of a $K_4$ with a pendant edge.

math.CO↗

arXiv2017

Vertex-disjoint properly edge-colored cycles in edge-colored complete graphs

It is conjectured that every edge-colored complete graph $G$ on $n$ vertices satisfying $Δ^{mon}(G)\leq n-3k+1$ contains $k$ vertex-disjoint properly edge-colored cycles. We confirm this conjecture for $k=2$, prove several additional weaker results for general $k$, and we establish structural properties of possible minimum counterexamples to the conjecture. We also reveal a close relationship between properly edge-colored cycles in edge-colored complete graphs and directed cycles in multi-partite tournaments. Using this relationship and our results on edge-colored complete graphs, we obtain several partial solutions to a conjecture on disjoint cycles in directed graphs due to Bermond and Thomassen.

math.CO↗

arXiv2017

Computability and Complexity of Unconventional Computing Devices

We discuss some claims that certain UCOMP devices can perform hypercomputation (compute Turing-uncomputable functions) or perform super-Turing computation (solve NP-complete problems in polynomial time). We discover that all these claims rely on the provision of one or more unphysical resources.

cs.ET↗

arXiv2013

Linear-Time Algorithms for Scattering Number and Hamilton-Connectivity of Interval Graphs

Hung and Chang showed that for all k>=1 an interval graph has a path cover of size at most k if and only if its scattering number is at most k. They also showed that an interval graph has a Hamilton cycle if and only if its scattering number is at most 0. We complete this characterization by proving that for all k<=-1 an interval graph is -(k+1)-Hamilton-connected if and only if its scattering number is at most k. We also give an O(m+n) time algorithm for computing the scattering number of an interval graph with n vertices an m edges, which improves the O(n^4) time bound of Kratsch, Kloks and Müller. As a consequence of our two results the maximum k for which an interval graph is k-Hamilton-connected can be computed in O(m+n) time.

cs.DS↗