Author

# Ryan Jeong

Bio: Ryan Jeong is an academic researcher. The author has contributed to research in topics: Mathematics. The author has an hindex of 2, co-authored 7 publications receiving 17 citations.

Topics: Mathematics

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19 Mar 2022

TL;DR: In this paper , the authors studied necessary and sufficient conditions for FS (X, Y ) to be connected for all graphs X from some set, and showed that FS ( X, Y) is connected if and only if Y is a forest with trees of jointly coprime size.

Abstract: Given two graphs X and Y with the same number of vertices, the friends-and-strangers graph FS ( X, Y ) has as its vertices all n ! bijections from V ( X ) to V ( Y ), with bijections σ, τ adjacent if and only if they diﬀer on two elements of V ( X ), whose mappings are adjacent in Y . In this article, we study necessary and suﬃcient conditions for FS ( X, Y ) to be connected for all graphs X from some set. In the setting that we take X to be drawn from the set of all biconnected graphs, we prove that FS ( X, Y ) is connected for all biconnected X if and only if Y is a forest with trees of jointly coprime size; this resolves a conjecture of Defant and Kravitz. We also initiate and make signiﬁcant progress toward determining the girth of FS ( X, Star n ) for connected graphs X , and in particular focus on the necessary trajectories that the central vertex of Star n takes around all such graphs X to achieve the girth.

9 citations

TL;DR: In this article , the diameters of connected components of friends-and-strangers graphs were shown to fail to be polynomially bounded in the size of X and Y, resolving a question raised by Alon, Defant, and Kravitz in the negative.

Abstract: Given two graphs X and Y on n vertices, the friends-and-strangers graph FS(X,Y ) has as its vertices all n! bijections from V (X) to V (Y ), with bijections σ, τ adjacent if and only if they differ on two elements of V (X), whose mappings are adjacent in Y . In this work, we study the diameters of friends-and-strangers graphs, which correspond to the largest number of swaps necessary to achieve one configuration from another. We provide families of constructions XL and YL for all integers L ≥ 1 to show that diameters of connected components of friends-and-strangers graphs fail to be polynomially bounded in the size of X and Y , resolving a question raised by Alon, Defant, and Kravitz in the negative. Specifically, our construction yields that there exist infinitely many values of n for which there are n-vertex graphs X and Y with the diameter of a component of FS(X,Y ) at least n . We also study the diameters of components of friends-and-strangers graphs when X is taken to be a path graph or a cycle graph, showing that any component of FS(Pathn, Y ) has diameter at most |E(Y )|, and diam(FS(Cyclen, Y )) is O(n) whenever FS(Cyclen, Y ) is connected. We conclude the work with several conjectures that aim to generalize this latter result.

8 citations

03 Oct 2022

TL;DR: For the generalized dihedral group D = Z 2 (cid:110) G, the authors showed that there are more MSTD sets than MDTS sets when 6 ≤ m ≤ c j √ n for c j = 1 . 3229 / √ 111 + 5 j , where j is the number of elements in G with order at most 2.

Abstract: . Given a group G , we say that a set A ⊆ G has more sums than diﬀerences (MSTD) if | A + A | > | A − A | , has more diﬀerences than sums (MDTS) if | A + A | < | A − A | , or is sum-diﬀerence balanced if | A + A | = | A − A | . A problem of recent interest has been to understand the frequencies of these type of subsets. The seventh author and Vissuet studied the problem for arbitrary ﬁnite groups G and proved that almost all subsets A ⊆ G are sum-diﬀerence balanced as | G | → ∞ . For the dihedral group D 2 n , they conjectured that of the remaining sets, most are MSTD, i.e., there are more MSTD sets than MDTS sets. Some progress on this conjecture was made by Haviland et al. in 2020, when they introduced the idea of partitioning the subsets by size: if, for each m , there are more MSTD subsets of D 2 n of size m than MDTS subsets of size m , then the conjecture follows. We extend the conjecture to generalized dihedral groups D = Z 2 (cid:110) G , where G is an abelian group of size n and the nonidentity element of Z 2 acts by inversion. We make further progress on the conjecture by considering subsets with a ﬁxed number of rotations and reﬂections. By bounding the expected number of overlapping sums, we show that the collection S D,m of subsets of the generalized dihedral group D of size m has more MSTD sets than MDTS sets when 6 ≤ m ≤ c j √ n for c j = 1 . 3229 / √ 111 + 5 j , where j is the number of elements in G with order at most 2. We also analyze the expectation for | A + A | and | A − A | for A ⊆ D 2 n , proving an explicit formula for | A − A | when n is prime.

03 Jan 2022

TL;DR: In this article , the authors studied the diameters of connected components of friends-and-strangers graphs and showed that the diameter of a component corresponds to the largest number of swaps necessary to go from one configuration in the component to another.

Abstract: Given $n$-vertex simple graphs $X$ and $Y$, the friends-and-strangers graph $\mathsf{FS}(X, Y)$ has as its vertices all $n!$ bijections from $V(X)$ to $V(Y)$, with bijections $\sigma, \tau$ adjacent if and only if they differ on two adjacent elements of $V(X)$ with mappings adjacent in $Y$. We study the diameters of connected components of friends-and-strangers graphs: the diameter of a component of $\mathsf{FS}(X,Y)$ corresponds to the largest number of swaps necessary to go from one configuration in the component to another. We study the diameters of components of friends-and-strangers graphs when fixing $X$ to be a particular kind of graph, showing that any component of $\mathsf{FS}(\mathsf{Path}_n, Y)$ has $O(n^2)$ diameter and that any component of $\mathsf{FS}(\mathsf{Cycle}_n, Y)$ has $O(n^4)$ diameter, improvable to $O(n^3)$ whenever $\mathsf{FS}(\mathsf{Cycle}_n, Y)$ is connected. Using an explicit construction, we show that for all $n \geq 2$, there exist $n$-vertex graphs $X$ and $Y$ such that $\mathsf{FS}(X,Y)$ has a component with diameter at least $e^{\Omega(n)}$. This resolves a question raised by Alon, Defant, and Kravitz in the negative. We conclude the work with several suggestions for future research, including a conjecture which asserts that connected friends-and-strangers graphs always have polynomial diameter in $n$.

20 Oct 2022

TL;DR: In this paper , the authors studied the probabilistic aspects of random Zeckendorf games and showed that for any input N , the range of possible game lengths constitutes an interval of natural numbers: every game length between the shortest and longest game lengths can be achieved.

Abstract: . Zeckendorf proved that any positive integer has a unique decomposition as a sum of non-consecutive Fibonacci numbers, indexed by F 1 = 1 , F 2 = 2 , F n +1 = F n + F n − 1 . Motivated by this result, Baird, Epstein, Flint, and Miller [3] deﬁned the two-player Zeckendorf game, where two players take turns acting on a multiset of Fibonacci numbers that always sums to N . The game terminates when no possible moves remain, and the ﬁnal player to perform a move wins. Notably, [3] studied the setting of random games: the game proceeds by choosing an available move uniformly at random, and they conjecture that as the input N → ∞ , the distribution of random game lengths converges to a Gaussian. We prove that certain sums of move counts is constant, and ﬁnd a lower bound on the number of shortest games on input N involving the Catalan numbers. The works [3] and Cuzensa et al. [5] determined how to achieve a shortest and longest possible Zeckendorf game on a given input N , respectively: we establish that for any input N , the range of possible game lengths constitutes an interval of natural numbers: every game length between the shortest and longest game lengths can be achieved. We further the study of probabilistic aspects of random Zeckendorf games. We study two probability measures on the space of all Zeckendorf games on input N : the uniform measure, and the measure induced by choosing moves uniformly at random at any given position. Under both measures that in the limit N → ∞ , both players win with probability 1 / 2. We also ﬁnd natural partitions of the collection of all Zeckendorf games of a ﬁxed input N , on which we observe weak convergence to a Gaussian in the limit N → ∞ . We conclude the work with many open problems.

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19 Mar 2022

TL;DR: In this paper , the authors studied necessary and sufficient conditions for FS (X, Y ) to be connected for all graphs X from some set, and showed that FS ( X, Y) is connected if and only if Y is a forest with trees of jointly coprime size.

Abstract: Given two graphs X and Y with the same number of vertices, the friends-and-strangers graph FS ( X, Y ) has as its vertices all n ! bijections from V ( X ) to V ( Y ), with bijections σ, τ adjacent if and only if they diﬀer on two elements of V ( X ), whose mappings are adjacent in Y . In this article, we study necessary and suﬃcient conditions for FS ( X, Y ) to be connected for all graphs X from some set. In the setting that we take X to be drawn from the set of all biconnected graphs, we prove that FS ( X, Y ) is connected for all biconnected X if and only if Y is a forest with trees of jointly coprime size; this resolves a conjecture of Defant and Kravitz. We also initiate and make signiﬁcant progress toward determining the girth of FS ( X, Star n ) for connected graphs X , and in particular focus on the necessary trajectories that the central vertex of Star n takes around all such graphs X to achieve the girth.

9 citations

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TL;DR: The cycle spaces of friends-and-strangers graphs were studied in this article , where the authors gave necessary and sufficient conditions for the cycle spaces to be connected in the case k = 3.

Abstract: If $X=(V(X),E(X))$ and $Y=(V(Y),E(Y))$ are $n$-vertex graphs, then their friends-and-strangers graph $\mathsf{FS}(X,Y)$ is the graph whose vertices are the bijections from $V(X)$ to $V(Y)$ in which two bijections $\sigma$ and $\sigma'$ are adjacent if and only if there is an edge $\{a,b\}\in E(X)$ such that $\{\sigma(a),\sigma(b)\}\in E(Y)$ and $\sigma'=\sigma\circ (a\,\,b)$, where $(a\,\,b)$ is the permutation of $V(X)$ that swaps $a$ and $b$. We prove general theorems that provide necessary and/or sufficient conditions for $\mathsf{FS}(X,Y)$ to be connected. As a corollary, we obtain a complete characterization of the graphs $Y$ such that $\mathsf{FS}(\mathsf{Dand}_{k,n},Y)$ is connected, where $\mathsf{Dand}_{k,n}$ is a dandelion graph; this substantially generalizes a theorem of the first author and Kravitz in the case $k=3$. For specific choices of $Y$, we characterize the spider graphs $X$ such that $\mathsf{FS}(X,Y)$ is connected. In a different vein, we study the cycle spaces of friends-and-strangers graphs. Naatz proved that if $X$ is a path graph, then the cycle space of $\mathsf{FS}(X,Y)$ is spanned by $4$-cycles and $6$-cycles; we show that the same statement holds when $X$ is a cycle and $Y$ has domination number at least $3$. When $X$ is a cycle and $Y$ has domination number at least $2$, our proof sheds light on how walks in $\mathsf{FS}(X,Y)$ behave under certain Coxeter moves.

9 citations

TL;DR: In this article , the diameters of connected components of friends-and-strangers graphs were shown to fail to be polynomially bounded in the size of X and Y, resolving a question raised by Alon, Defant, and Kravitz in the negative.

Abstract: Given two graphs X and Y on n vertices, the friends-and-strangers graph FS(X,Y ) has as its vertices all n! bijections from V (X) to V (Y ), with bijections σ, τ adjacent if and only if they differ on two elements of V (X), whose mappings are adjacent in Y . In this work, we study the diameters of friends-and-strangers graphs, which correspond to the largest number of swaps necessary to achieve one configuration from another. We provide families of constructions XL and YL for all integers L ≥ 1 to show that diameters of connected components of friends-and-strangers graphs fail to be polynomially bounded in the size of X and Y , resolving a question raised by Alon, Defant, and Kravitz in the negative. Specifically, our construction yields that there exist infinitely many values of n for which there are n-vertex graphs X and Y with the diameter of a component of FS(X,Y ) at least n . We also study the diameters of components of friends-and-strangers graphs when X is taken to be a path graph or a cycle graph, showing that any component of FS(Pathn, Y ) has diameter at most |E(Y )|, and diam(FS(Cyclen, Y )) is O(n) whenever FS(Cyclen, Y ) is connected. We conclude the work with several conjectures that aim to generalize this latter result.

8 citations

••

TL;DR: In this article , it was shown that if p1p2≥p02=n−1/2+o(1) and p1,p2 ≥w(n)p0, where w (n)→0 as n→∞, then FS(X,Y) is connected with high probability, which extends the result on p1=p2=p, due to Alon, Defant and Kravitz.

8 citations

08 Oct 2022

TL;DR: In this article , the connectivity of friends-and-strangers graphs arising from two random bipartite graphs is investigated and a tight bound, up to a factor of n o (1) , for the threshold probability at which such graphs attain maximal connectivity is obtained.

Abstract: . In this paper, we investigate the connectivity of friends-and-strangers graphs, which were introduced by Defant and Kravitz in 2020. We begin by considering friends-and-strangers graphs arising from two random bipartite graphs, independently chosen from G ( K n,n ,p ), and we obtain a tight bound, up to a factor of n o (1) , for the threshold probability at which such graphs attain maximal connectivity. This resolves a conjecture of Alon, Defant, and Kravitz up to lower order terms. Further, we introduce a generalization of the notion of friends-and-strangers graphs in which vertices of the starting graphs are allowed to have multiplicities and obtain generalizations of previous results of Wilson and of Defant and Kravitz in this new setting.

6 citations