Year: 2025
Type: article
Abstract: ABSTRACT Given a graph , the unlabeled subgraphs are called the cards of . The deck of is the multiset . Wendy Myrvold showed that a disconnected graph and a connected graph both on vertices have at most cards in common and found (infinite) families of trees and disconnected forests for which this upper bound is tight. Bowler, Brown, and Fenner conjectured that this bound is tight for . In this article, we prove this conjecture for sufficiently large . The main result is that a tree and a unicyclic graph on vertices have at most common cards. Combined with Myrvold's work, this shows that it can be determined whether a graph on vertices is a tree from any of its cards. On the basis of this theorem, it follows that any forest and nonforest also have at most common cards. Furthermore, the main ideas of the proof for trees are used to show that the girth of a graph on vertices can be determined based on any of its cards. Lastly, we show that any cards determine whether a graph is bipartite.
Source: Journal of Graph Theory
DOI: 10.1002/jgt.23274
Author: Gabriëlle Zwaneveld
Institution: University of Amsterdam
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $4,150
OpenAlex ID: W4411905942