An updated survey on rainbow connections of graphs - a dynamic survey

The concept of rainbow connection was introduced by Chartrand, Johns, McKeon and Zhang in 2008. Nowadays it has become a new and active subject in graph theory. There is a book on this topic by Li and Sun in 2012, and a survey paper by Li, Shi and Sun in 2013. More and more researchers are working in this field, and many new papers have been published in journals. In this survey we attempt to bring together most of the new results and papers that deal with this topic. We begin with an introduction, and then try to organize the work into the following categories, rainbow connection coloring of edge-version, rainbow connection coloring of vertex-version, rainbow $k$-connectivity, rainbow index, rainbow connection coloring of total-version, rainbow connection on digraphs, rainbow connection on hypergraphs. This survey also contains some conjectures, open problems and questions for further study

Essentially tight bounds for rainbow cycles in proper edge-colourings

An edge-coloured graph is said to be rainbow if no colour appears more than once. Extremal problems involving rainbow objects have been a focus of much research over the last decade as they capture the essence of a number of interesting problems in a variety of areas. A particularly intensively studied question due to Keevash, Mubayi, Sudakov and Verstra\"ete from 2007 asks for the maximum possible average degree of a properly edge-coloured graph on $n$ vertices without a rainbow cycle. Improving upon a series of earlier bounds, Tomon proved an upper bound of $(\log n)^{2+o(1)}$ for this question. Very recently, Janzer-Sudakov and Kim-Lee-Liu-Tran independently removed the $o(1)$ term in Tomon's bound, showing a bound of $O(\log^2 n)$. We prove an upper bound of $(\log n)^{1+o(1)}$ for this maximum possible average degree when there is no rainbow cycle. Our result is tight up to the $o(1)$ term, and so it essentially resolves this question. In addition, we observe a connection between this problem and several questions in additive number theory, allowing us to extend existing results on these questions for abelian groups to the case of non-abelian groups

Indeks jumlah jarak eksentrik graf invers dari grup quaternion diperumum

INDONESIA: Graf invers dari suatu grup adalah graf yang himpunan titiknya adalah semua anggota grup berhingga sedemikian sehingga dua titik berbeda u dan v terhubung langsung jika hanya jika u∙v∈S atau v∙u∈S. Grup quaternion diperumum adalah grup non-abelian dengan orde 4n yang dibangun oleh dua elemen a,b yang disimbolkan 〈a,b〉 didefinisikan dengan Q_4=⟨a.b│a^2n=e,b^2=a^n,b∙a∙b^(-1)=a^(-1) ⟩ dan e merupakan elemen identitas, n≥N dan n≥2. Penelitian ini bertujuan untuk mengetahui formula indeks jumlah eksentrik graf invers dari grup quaternion diperumum dengan n≥N dan n≥2. Penelitian ini difokuskan pada formula umum indeks jumlah eksentrik graf invers dari grup quaternion diperumum dengan n≥N dan n≥2. Hasil dari penelitian ini adalah formula indeks jumlah jarak eksentrik graf invers dari grup quaternion diperumum. ENGLISH: The inverse graph of a group is a graph which the vertices set are all elements of a finite group and two distinct vertices u and v are adjacent if and only if either u∙v∈S or v∙u∈S. A generalized quaternion group is a non-abelian group with order 4n constructed from the two elements a,b symbolized 〈a,b〉 defined by Q_4=⟨a.b│a^2n=e,b^2=a^n,b∙a∙b^(-1)=a^(-1)⟩ and the e is identity, n≥N and n≥2. The purpose of this research are to find out the eccentric distance sum index formula of inverse graph of a generalized quaternion group with n≥N and n≥2. The study focused on the general formula of inverse graph of a generalized quaternion group with n≥N and n≥2. The result of this research is the eccentric distance sum index formula of inverse graph of a generalized quaternion group. ARABIC: المخطاط العكسي من الزمرة هوللمخطاط له مجموعة من الرؤوس هي جميع أعضاء زمرة المحدودة بحيث ترتبط رأسان مختلفان u و v مباشرة إذا وفقط أذ u∙v∈S أو v∙u∈S. زمرة Quaternion المعممة هي زمرة غيربليانية (non-abelian) بنسق4n التي مكون بواسطة عنصرين a,b يرمز اليه 〈a,b〉 يمكن تعريفها بـ Q_4=⟨a∙b│a^2n=e,b^2=a^n,b∙a∙b^(-1)=a^(-1)⟩ و e الوحدة، n≥N و n≥2. الأغراض في هذه المعرفة الصيغ Eccentric Distance Sum Index للمخطاط المعكوس من زمرة Quaternion المعممة بـ n≥N و n≥2. يركز هذا البحث على الصيغة العامة Eccentric Distance Sum Index للمخطاط المعكوس من زمرة Quaternion المعممة بـ n≥N و n≥2. وأما نتائج هذا البحث هيEccentric Distance Sum Index للمخطاط المعكوس من زمرة Quaternion المعممة

Combinatorics, Probability and Computing

The main theme of this workshop was the use of probabilistic methods in combinatorics and theoretical computer science. Although these methods have been around for decades, they are being refined all the time: they are getting more and more sophisticated and powerful. Another theme was the study of random combinatorial structures, either for their own sake, or to tackle extremal questions. The workshop also emphasized connections between probabilistic combinatorics and discrete probability

Applications of Geometric and Spectral Methods in Graph Theory

Networks, or graphs, are useful for studying many things in today’s world. Graphs can be used to represent connections on social media, transportation networks, or even the internet. Because of this, it’s helpful to study graphs and learn what we can say about the structure of a given graph or what properties it might have. This dissertation focuses on the use of the probabilistic method and spectral graph theory to understand the geometric structure of graphs and find structures in graphs. We will also discuss graph curvature and how curvature lower bounds can be used to give us information about properties of graphs. A rainbow spanning tree in an edge-colored graph is a spanning tree in which each edge is a different color. Carraher, Hartke, and Horn showed that for n and C large enough, if G is an edge-colored copy of Kn in which each color class has size at most n/2, then G has at least [n/(C log n)] edge-disjoint rainbow spanning trees. Here we show that spectral graph theory can be used to prove that if G is any edge-colored graph with n vertices in which each color appears on at most δλ1/2 edges, where δ ≥ C log n for n and C sufficiently large and λ1 is the second-smallest eigenvalue of the normalized Laplacian matrix of G, then G contains at least [δλ1/ C log n] edge-disjoint rainbow spanning trees. We show how curvature lower bounds can be used in the context of understanding (personalized) PageRank, which was developed by Brin and Page. PageRank ranks the importance of webpages near a seed webpage, and we are interested in how this importance diffuses. We do this by using a notion of graph curvature introduced by Bauer, Horn, Lin, Lippner, Mangoubi, and Yau