Two-connected graphs with prescribed three-connected components
Andrei Gagarin, Gilbert Labelle, Pierre Leroux, Timothy Walsh
备注：
Work presented at the Ottawa-Carleton Discrete Mathematics Workshop, May 25-26, 2007 and at the Seminaire Lotharingien de Combinatoire, Bertinoro, Italy, September 24-26, 2007. 32 pages. 11 pdf figures. Version 2: Minor revisions, one Table added
We adapt the classical 3-decomposition of any 2-connected graph to the case of simple graphs (no loops or multiple edges). By analogy with the block-cutpoint tree of a connected graph, we deduce from this decomposition a bicolored tree tc(g) associated with any 2-connected graph g, whose white vertices are the 3-components of g (3-connected components or polygons) and whose black vertices are bonds linking together these 3-components, arising from separating pairs of vertices of g. Two fundamental relationships on graphs and networks follow from this construction. The first one is a dissymmetry theorem which leads to the expression of the class B=B(F) of 2-connected graphs, all of whose 3-connected components belong to a given class F of 3-connected graphs, in terms of various rootings of B. The second one is a functional equation which characterizes the corresponding class R=R(F) of two-pole networks all of whose 3-connected components are in F. All the rootings of B are then expressed in terms of F and R. There follow corresponding identities for all the associated series, in particular the edge index series. Numerous enumerative consequences are discussed.
On the maximum spectral radius of connected graphs with a prescribed order and size
Faculty of Electronic Engineering, University of Niš（尼什大学电子工程学院）；Faculty of Mathematics, Natural Sciences and Information Technologies, University of Primorska（滨海大学数学、自然科学和信息技术学院）
Nearest Graph Laplacians with Prescribed Connected Components: A Convex Framework for Network Reconstruction
Indian Institute of Science Education and Research, Bhopal（印度科学教育研究所博帕尔分校）；Shiv Nadar Institution of Eminence, Delhi-NCR（希夫·纳德卓越机构德里NCR校区）；Indian Institute of Technology Guwahati（印度理工学院古瓦哈提分校）
Low chromatic spanning sub(di)graphs with prescribed degree or connectivity properties
推广了Erdés和Lovász的著名结果，证明了每个图$G$都包含一个具有$λ(H)\geq \lceil{}\frac{k-1}{k}λ(G)\rceil$的生成$k$ -部子图$H$，其中$λ(G)$是$G$的边连通度.特别是，连同一个众所周知的结果，由于纳什-威廉姆斯和Tutte，这意味着每一个$7$ -边连通图包含一个生成二部图，其边集分解成两个边不相交的生成树。我们证明了这是最好的可能性，因为它不适用于无限多的$6$ -边连通图。对于有向图，文献[6]证明了不存在$k$使得每个$k$ -弧连通有向图都有一个生成强二部子有向图.证明了每个强有向图都有一个生成强三部子有向图，并且每个至少6个顶点的强半完全有向图都有一个生成强二部子有向图。\jbj{We generalize this result to higher connectivities by proving}证明了，对于每个正整数$k$，每个$k$ -弧连通有向图包含一个生成$(2k+1$）-有部子图是$k$ -弧连通的，这是最佳可能的.文[18]中的一个猜想是：每个最小出度$2k-1$的有向图都包含一个最小出度至少为$k$的生成$3$ -部子有向图.通过给出一类具有最小出度$2k-2$的有向图，证明了有界$2k-1$是最佳可能的，这类有向图不包含任何出度至少为$k$的生成$3$ -部子有向图.证明了最小半度至少为…
On degree sum conditions for 2-factors with a prescribed number of cycles
对于图$G$的顶点子集$X$，设$Δ_{t}(X)$是大小为$t$的$X$的子集的度和的最大值。本文证明了如下结果：设$k$为正整数，$G$为$n \ge 5k - 2$阶的$m$ -连通图.如果对于$G$中大小为$\lceil m/k \rceil+1$的每个独立集合$X$，$Δ_{2}(X) \ge n$，则$G$具有恰好为$k$循环的2-因子。这是Brandt等人[Degree conditions for 2-factors，J. Graph Theory 24（1997）165-173]和Yamashita [On degree sum conditions for long cycles and cycles through specified vertices，Discrete Math.308（2008）6584-6587]分别获得的结果的共同推广。
