arXiv subjects

Lale Özkahya

Publications and source records attributed to Lale Özkahya.

10 recordsLinked to original sources

arXiv2026

Star Coloring of Hypergraphs

We study a generalization of the star coloring problem on hypergraphs. For a family of connected subhypergraphs $\mathcal{F}$, we define an $\mathcal{F}$-coloring of a hypergraph as a coloring avoiding monochromatic hyperedges and any 2-colored member of $\mathcal{F}$. We let $χ^r_{\mathcal{F}}(d)$ be the maximum of the minimum number of colors needed for an $\mathcal{F}$-coloring of an $r$-uniform hypergraph with maximum degree $d.$ We show bounds for $χ^r_{\mathcal{F}}(d)$, that also yield results on star and acyclic coloring problem on hypergraphs.

math.CO↗

arXiv2023

Coloring of Graphs Avoiding Bicolored Paths of a Fixed Length

The problem of finding the minimum number of colors to color a graph properly without containing any bicolored copy of a fixed family of subgraphs has been widely studied. Most well-known examples are star coloring and acyclic coloring of graphs (Grünbaum, 1973) where bicolored copies of $P_4$ and cycles are not allowed, respectively. In this paper, we introduce a variation of these problems and study proper coloring of graphs not containing a bicolored path of a fixed length and provide general bounds for all graphs. A $P_k$-coloring of an undirected graph $G$ is a proper vertex coloring of $G$ such that there is no bicolored copy of $P_k$ in $G,$ and the minimum number of colors needed for a $P_k$-coloring of $G$ is called the $P_k$-chromatic number of $G,$ denoted by $s_k(G).$ We provide bounds on $s_k(G)$ for all graphs, in particular, proving that for any graph $G$ with maximum degree $d\geq 2,$ and $k\geq4,$ $s_k(G)=O(d^{\frac{k-1}{k-2}}).$ Moreover, we find the exact values for the $P_k$-chromatic number of the products of some cycles and paths for $k=5,6.$

math.CO↗

arXiv2022

Learning Heuristics for the Maximum Clique Enumeration Problem Using Low Dimensional Representations

Approximate solutions to various NP-hard combinatorial optimization problems have been found by learned heuristics using complex learning models. In particular, vertex (node) classification in graphs has been a helpful method towards finding the decision boundary to distinguish vertices in an optimal set from the rest. By following this approach, we use a learning framework for a pruning process of the input graph towards reducing the runtime of the maximum clique enumeration problem. We extensively study the role of using different vertex representations on the performance of this heuristic method, using graph embedding algorithms, such as Node2vec and DeepWalk, and representations using higher-order graph features comprising local subgraph counts. Our results show that Node2Vec and DeepWalk are promising embedding methods in representing nodes towards classification purposes. We observe that using local graph features in the classification process produce more accurate results when combined with a feature elimination process. Finally, we provide tests on random graphs to show the robustness and scalability of our method.

cs.SI↗

arXiv2022

Low-Rank Representations Towards Classification Problem of Complex Networks

Complex networks representing social interactions, brain activities, molecular structures have been studied widely to be able to understand and predict their characteristics as graphs. Models and algorithms for these networks are used in real-life applications, such as search engines, and recommender systems. In general, such networks are modelled by constructing a low-dimensional Euclidean embedding of the vertices of the network, where proximity of the vertices in the Euclidean space hints the likelihood of an edge (link). In this work, we study the performance of such low-rank representations of real-life networks on a network classification problem.

cs.SI↗

arXiv2020

A Fast Counting Method for 6-motifs with Low Connectivity

A $k$-motif (or graphlet) is a subgraph on $k$ nodes in a graph or network. Counting of motifs in complex networks has been a well-studied problem in network analysis of various real-word graphs arising from the study of social networks and bioinformatics. In particular, the triangle counting problem has received much attention due to its significance in understanding the behavior of social networks. Similarly, subgraphs with more than 3 nodes have received much attention recently. While there have been successful methods developed on this problem, most of the existing algorithms are not scalable to large networks with millions of nodes and edges. The main contribution of this paper is a preliminary study that genaralizes the exact counting algorithm provided by Pinar, Seshadhri and Vishal to a collection of 6-motifs. This method uses the counts of motifs with smaller size to obtain the counts of 6-motifs with low connecivity, that is, containing a cut-vertex or a cut-edge. Therefore, it circumvents the combinatorial explosion that naturally arises when counting subgraphs in large networks.

cs.SI↗

arXiv2017

Minimum Rainbow $H$-Decompositions of Graphs

Given graphs $G$ and $H$, we consider the problem of decomposing a properly edge-colored graph $G$ into few parts consisting of rainbow copies of $H$ and single edges. We establish a close relation to the previously studied problem of minimum $H$-decompositions, where an edge coloring does not matter and one is merely interested in decomposing graphs into copies of $H$ and single edges.

math.CO↗

arXiv2014

On 3-uniform hypergraphs without a cycle of a given length

We study the maximum number of hyperedges in a 3-uniform hypergraph on $n$ vertices that does not contain a Berge cycle of a given length $\ell$. In particular we prove that the upper bound for $C_{2k+1}$-free hypergraphs is of the order $O(k^2n^{1+1/k})$, improving the upper bound of Györi and Lemons by a factor of $Θ(k^2)$. Similar bounds are shown for linear hypergraphs.

math.CO↗

arXiv2012

On homometric sets in graphs

For a vertex set $S\subseteq V(G)$ in a graph $G$, the {\em distance multiset}, $D(S)$, is the multiset of pairwise distances between vertices of $S$ in $G$. Two vertex sets are called {\em homometric} if their distance multisets are identical. For a graph $G$, the largest integer $h$, such that there are two disjoint homometric sets of order $h$ in $G$, is denoted by $h(G)$. We slightly improve the general bound on this parameter introduced by Albertson, Pach and Young (2010) and investigate it in more detail for trees and graphs of bounded diameter. In particular, we show that for any tree $T$ on $n$ vertices $h(T) \geq \sqrt[3]{n}$ and for any graph $G$ of fixed diameter $d$, $h(G) \geq cn^{1/ (2d-2)}$.

math.CO↗

arXiv2011

On a covering problem in the hypercube

In this paper, we address a particular variation of the Turán problem for the hypercube. Alon, Krech and Szabó (2007) asked "In an n-dimensional hypercube, Qn, and for l < d < n, what is the size of a smallest set, S, of Q_l's so that every Q_d contains at least one member of S?" Likewise, they asked a similar Ramsey type question: "What is the largest number of colors that we can use to color the copies of Q_l in Q_n such that each Q_d contains a Q_l of each color?" We give upper and lower bounds for each of these questions and provide constructions of the set S above for some specific cases.

math.CO↗

arXiv2011

On the ratio of maximum and minimum degree in maximal intersecting families

To study how balanced or unbalanced a maximal intersecting family $\mathcal{F}\subseteq \binom{[n]}{r}$ is we consider the ratio $\mathcal{R}(\mathcal{F})=\frac{Δ(\mathcal{F})}{δ(\mathcal{F})}$ of its maximum and minimum degree. We determine the order of magnitude of the function $m(n,r)$, the minimum possible value of $\mathcal{R}(\mathcal{F})$, and establish some lower and upper bounds on the function $M(n,r)$, the maximum possible value of $\mathcal{R}(\mathcal{F})$. To obtain constructions that show the bounds on $m(n,r)$ we use a theorem of Blokhuis on the minimum size of a non-trivial blocking set in projective planes.

math.CO↗