arXiv subjects

András London

Publications and source records attributed to András London.

6 recordsLinked to original sources

arXiv2026

Spanning-Tree Extremality in $C_4$-Free Graphs

We study the maximum number of spanning trees in connected $n$-vertex $C_4$-free graphs. For projective-plane orders $n=q^2+q+1$, we determine the spanning-tree count of every polarity graph and show that polarity graphs with exactly $q+1$ absolute points maximize this count within the polarity family, attaining $n^{(n-3)/2}$ spanning trees. Combined with a stability theorem of He, Ma and Yang \cite{HeMaYangCSIAM23}, this yields the same exact upper bound for all sufficiently dense $C_4$-free graphs in the known polarity stability regime. For arbitrary $C_4$-free graphs at these orders, we derive a global upper bound implying \[ \log \mathrm{st}(n,C_4)=\frac{n-3}{2}\log n+O(\sqrt n), \] where $\mathrm{st}(n,C_4)$ denotes the maximum number of spanning trees over connected $n$-vertex $C_4$-free graphs.

math.CO↗

arXiv2026

A Degree-Preserving Builder--Chooser Game

We propose a degree-preserving variant of the Builder--Chooser clique game of Pettie, Tardos, and Walczak. In each round, Builder chooses a matching, performs a degree-preserving growth (DPG) step by replacing the chosen edges with edges incident to a new vertex. Then partitions the entire edge set into two parts, and Chooser keeps one part. We begin the study of this game with the first nontrivial target, namely forcing a triangle. For triangle-free initial graphs we prove an exact one-round criterion, derive an exact one-round threshold on paths and exact forcing times on cycles, and identify the $5$-cycle as the first genuine two-round example. We then formulate a one-round criterion for larger cliques, prove a sharp exact result for forcing $K_4$ from triangle-free seeds. We establish general lower bounds on clique-forcing times from clique-free seeds, and isolate a conjectural template-amplifier lemma which, if proved, would imply that every clique is forceable from some triangle-free seed.

math.CO↗

arXiv2026

Edge partitions into induced-$2K_2$-free bipartite graphs

We study edge partitions of a bipartite graph into induced-$2K_2$-free bipartite graphs, i.e.\ into Ferrers (chain) graphs. We define $\fp(G)$ as the minimum number of parts in such a partition. We prove general lower and upper bounds in terms of induced matchings and Dilworth widths of neighborhood posets. We compute the parameter exactly for paths and even cycles, and we exhibit separations showing that the induced-matching lower bound and the width upper bound can both be far from tight. We also record a simple host-induced conflict-graph lower bound, present a $0$--$1$ matrix viewpoint, and add some complexity remarks.

math.CO↗

arXiv2025

Partitions of Graphs into Special Bipartite Graphs

We study the problem of partitioning the edge set of the complete graph into bipartite subgraphs under certain constraints defined by forbidden subgraphs. These constraints lead to both classical problems, such as partitioning into independent matchings or complete bipartite subgraphs, and novel variants motivated by structural restrictions. Our theoretical framework is inspired by clustering problems in real-world transaction graphs, which can be formulated naturally as edge partitioning problems under bipartite graph constraints. The main result of this paper is the proof of the bounds for $χ'_{2K_2}(n)$, which corresponds to the minimum number of induced $2K_2$-free bipartite subgraphs needed to partition the edges of $K_n$. In addition to this central result, we also present several similar bounds for other forbidden subgraphs on three or four vertices. Some are included primarily for the sake of completeness, to demonstrate the broad applicability of our approach, and some lead to other novel or well-known graph theoretical problems.

math.CO↗

arXiv2021

Graph clustering via generalized colorings

We propose a new approach for defining and searching clusters in graphs that represent real technological or transaction networks. In contrast to the standard way of finding dense parts of a graph, we concentrate on the structure of edges between the clusters, as it is motivated by some earlier observations, e.g. in the structure of networks in ecology and economics and by applications of discrete tomography. Mathematically special colorings and chromatic numbers of graphs are studied.

math.CO↗

arXiv2017

Core of communities in bipartite networks

We use the information present in a bipartite network to detect cores of communities of each set of the bipartite system. Cores of communities are found by investigating statistically validated projected networks obtained using information present in the bipartite network. Cores of communities are highly informative and robust with respect to the presence of errors or missing entries in the bipartite network. We assess the statistical robustness of cores by investigating an artificial benchmark network, the co-authorship network, and the actor-movie network. The accuracy and precision of the partition obtained with respect to the reference partition are measured in terms of the adjusted Rand index and of the adjusted Wallace index respectively. The detection of cores is highly precise although the accuracy of the methodology can be limited in some cases.

physics.soc-ph↗