Perfect 1‐Factorizations of a Family of Cayley Graphs

A 1‐factorization of a graph G is a decomposition of G into edge‐disjoint 1‐factors (perfect matchings), and a perfect 1‐factorization is a 1‐factorization in which the union of any two of the 1‐factors is a Hamilton cycle. We consider the problem of the existence of perfect 1‐factorizations of even order 4‐regular Cayley graphs, with a particular interest in circulant graphs. In this paper, we study a new family of graphs, denoted , which are Cayley graphs if and only if k is even or . By solving the perfect 1‐factorization problem for a large class of graphs, we obtain a new class of 4‐regular bipartite circulant graphs that do not have a perfect 1‐factorization, answering a problem posed in 7 . With further study of graphs, we prove the nonexistence of P1Fs in a class of 4‐regular non‐bipartite circulant graphs, which is further support for a conjecture made in 7 .     

Prediction of Compound Synthesis Accessibility Based on Reaction Knowledge Graph

With the increasing application of deep-learning-based generative models for de novo molecule design, the quantitative estimation of molecular synthetic accessibility (SA) has become a crucial factor for prioritizing the structures generated from generative models. It is also useful for helping in the prioritization of hit/lead compounds and guiding retrosynthesis analysis. In this study, based on the USPTO and Pistachio reaction datasets, a chemical reaction network was constructed for the identification of the shortest reaction paths (SRP) needed to synthesize compounds, and different SRP cut-offs were then used as the threshold to distinguish a organic compound as either an easy-to-synthesize (ES) or hard-to-synthesize (HS) class. Two synthesis accessibility models (DNN-ECFP model and graph-based CMPNN model) were built using deep learning/machine learning algorithms. Compared to other existing synthesis accessibility scoring schemes, such as SYBA, SCScore, and SAScore, our results show that CMPNN (ROC AUC: 0.791) performs better than SYBA (ROC AUC: 0.76), albeit marginally, and outperforms SAScore and SCScore. Our prediction models based on historical reaction knowledge could be a potential tool for estimating molecule SA.     

A shuffling theorem for reflectively symmetric tilings

The author and Rohatgi recently proved a ‘shuffling theorem’ for doubly-dented hexagons. In particular, they showed that shuffling removed unit triangles along a horizontal axis in a hexagon changes the tiling number by only a simple multiplicative factor. In this paper, we consider a similar phenomenon for a symmetry class of tilings, namely, the reflectively symmetric tilings. We also prove several shuffling theorems for halved hexagons.     

On the imaginary parts of chromatic roots

While much attention has been directed to the maximum modulus and maximum real part of chromatic roots of graphs of order $n$ (ie, with $n$ vertices), relatively little is known about the maximum imaginary part of such graphs. We prove that the maximum imaginary part can grow linearly in the order of the graph. We also show that for any fixed $\text{◂+▸}p\in \left(0,1\right)$, almost every random graph $G$ in the Erdös-Rényi model has a nonreal root.     

Planar graphs without triangles adjacent to cycles of length from 4 to 7 are 3-colorable

It is known that planar graphs without cycles of length from 4 to 7 are 3-colorable (Borodin et al., 2005) [13] and that planar graphs in which no triangles have common edges with cycles of length from 4 to 9 are 3-colorable (Borodin et al., 2006) [11]. We give a common extension of these results by proving that every planar graph in which no triangles have common edges with k-cycles, where k∈{4,5,7} (or, which is equivalent, with cycles of length 3, 5 and 7), is 3-colorable.     

Long cycles and paths in distance graphs

For n∈N and D⊆N, the distance graph has vertex set {0,1,…,n−1} and edge set {ij∣0≤i,j≤n−1,|j−i|∈D}. Note that the important and very well-studied circulant graphs coincide with the regular distance graphs.A fundamental result concerning circulant graphs is that for these graphs, a simple greatest common divisor condition, their connectivity, and the existence of a Hamiltonian cycle are all equivalent. Our main result suitably extends this equivalence to distance graphs. We prove that for a finite set D of order at least 2, there is a constant c_D such that the greatest common divisor of the integers in D is 1 if and only if for every n, has a component of order at least n−c_D if and only if for every n≥c_D+3, has a cycle of order at least n−c_D. Furthermore, we discuss some consequences and variants of this result.     

一个关于图是分数(k,n)-临界的邻域并条件

设G是一个图,以及k是满足1≤k的整数.一个图G在删除任意n个顶点后的子图均含有分数k-因子,则称G是一个分数(k,n)-临界图.给出了图是一个分数(k,n)-临界图的一个邻域并条件,并且该条件是最佳的.     

Set to set broadcasting in communication networks

Suppose G = (V,E) is a graph whose vertices represent people and edges represent telephone lines between pairs of people. Each person knows a unique message and is ignorant of the messages of other people at the beginning. These messages are then spread by telephone calls. In each call, two people exchange all information they have so far in exactly one unit of time. Suppose A and B are two nonempty subsets of V. The main purpose of this paper is to study the minimum number b(A,B,G) of telephone calls by which A broadcasts to B; and the minimum time t(A,B,G) such that A broadcasts to B. In particular, we give an exact formula for b(A,B,K_n) and linear-time algorithms for computing b(A,B,T) and t(A,B,T) of a tree T.     

Phase diagram of the Ising Model on a Cayley tree in the presence of competing interactions and magnetic field

We study the phase diagram for the Ising Model on a Cayley tree with competing nearest-neighbor interactionsJ ₁ and next-nearest-neighbor interactionsJ ₂ andJ ₃ in the presence of an external magnetic field. To perform this study, an iterative scheme similar to that appearing in real space renormalization group frameworks is established; it recovers, as particular cases, previous works by Vannimenus and by Inawashiroet al. At vanishing temperature, the phase diagram is fully determined, for all values and signs ofJ ₂/J ₁ andJ ₃/J ₂; in particular, we verify that values ofJ ₃/J ₂ high enough favor the paramagnetic phase. At finite temperatures, several interesting features (evolution of reentrances, separation of the modulated region into two disconnected pieces, etc.) are exhibited for typical values ofJ ₂/J ₁ andJ ₃/J ₂.Partially supported by the Brazilian Agencies CNPq and FINEP.