A New Method for Enumerating Independent Sets of a Fixed Size in General Graphs

We develop a new method for enumerating independent sets of a fixed size in general graphs, and we use this method to show that a conjecture of Engbers and Galvin [7] holds for all but finitely many graphs. We also use our method to prove special cases of a conjecture of Kahn [13]. In addition, we show that our method is particularly useful for computing the number of independent sets of small sizes in general regular graphs and Moore graphs, and we argue that it can be used in many other cases when dealing with graphs that have numerous structural restrictions.     

图的k部划分的一个注记

设k≥2是一个整数。本文证明了任意有m条边的图都存在一个顶点的划分V_1,V_2…,V_k,使得e(V_1,V_2…,V_k)≥k-1/k m+k-1/2k((2m+1/4)~1/2-1/2)-(k-2)~2/8k,且max{e(V_i):1≤i≤k}≤m/k~2+(k-1)/2k~2((2m+1/4)~1/2-1/2+3/8-7k-4/8k~2.我们的结果改进了[Fan G.,Hou J.,Zeng Q.,A bound for judicious k-partitions of graphs,Discrete Appl.Math.,2014,179:86—99]的主要结论.     

Large ‐ or ‐Minors in 3‐Connected Graphs

There are numerous results bounding the circumference of certain 3‐connected graphs. There is no good bound on the size of the largest bond (cocircuit) of a 3‐connected graph, however. Oporowski, Oxley, and Thomas (J Combin Theory Ser B 57 (1993), 2, 239–257) proved the following result in 1993. For every positive integer k, there is an integer such that every 3‐connected graph with at least n vertices contains a ‐ or ‐minor. This result implies that the size of the largest bond in a 3‐connected graph grows with the order of the graph. Oporowski et al. obtained a huge function iteratively. In this article, we first improve the above authors' result and provide a significantly smaller and simpler function . We then use the result to obtain a lower bound for the largest bond of a 3‐connected graph by showing that any 3‐connected graph on n vertices has a bond of size at least . In addition, we show the following: Let G be a 3‐connected planar or cubic graph on n vertices. Then for any , G has a ‐minor with , and thus a bond of size at least .     

Two‐sided Group Digraphs and Graphs

We study a family of digraphs (directed graphs) that generalises the class of Cayley digraphs. For nonempty subsets of a group G, we define the two‐sided group digraph to have vertex set G, and an arc from x to y if and only if for some and . In common with Cayley graphs and digraphs, two‐sided group digraphs may be useful to model networks as the same routing and communication scheme can be implemented at each vertex. We determine necessary and sufficient conditions on L and R under which may be viewed as a simple graph of valency , and we call such graphs two‐sided group graphs. We also give sufficient conditions for two‐sided group digraphs to be connected, vertex‐transitive, or Cayley graphs. Several open problems are posed. Many examples are given, including one on 12 vertices with connected components of sizes 4 and 8.     

A Structure Theorem for Strong Immersions

A graph H is strongly immersed in G if H is obtained from G by a sequence of vertex splittings (i.e., lifting some pairs of incident edges and removing the vertex) and edge removals. Equivalently, vertices of H are mapped to distinct vertices of G (branch vertices) and edges of H are mapped to pairwise edge‐disjoint paths in G, each of them joining the branch vertices corresponding to the ends of the edge and not containing any other branch vertices. We describe the structure of graphs avoiding a fixed graph as a strong immersion. The theorem roughly states that a graph which excludes a fixed graph as a strong immersion has a tree‐like decomposition into pieces glued together on small edge cuts such that each piece of the decomposition has a path‐like linear decomposition isolating the high degree vertices.     

Excluded Minors and the Ribbon Graphs of Knots

In this article we consider minors of ribbon graphs (or, equivalently, cellularly embedded graphs). The theory of minors of ribbon graphs differs from that of graphs in that contracting loops is necessary and doing this can create additional vertices and components. Thus, the ribbon graph minor relation is incompatible with the graph minor relation. We discuss excluded minor characterizations of minor closed families of ribbon graphs. Our main result is an excluded minor characterization of the family of ribbon graphs that represent knot and link diagrams.     

一个特殊6点图Q与nK_1,P_n及C_n的联图交叉数

图的交叉数是图的一个重要参数,研究图的交叉数问题是拓扑图论中的前沿难题.确定图的交叉数是NP-难问题,因为其难度,能够确定交叉数的图类很少.通过圆盘画法途径,确定了一个特殊6点图与n个孤立点nK_1,路P_n及圈C_n的联图的交叉数分别是cr(Q+nK_1)=Z(6,n)+2[n/2],cr(Q+P_n)=Z(6,n)+2[n/2]+1及cr(Q+C_n)=Z(6,n)+2[n/2]+3.