Looking for vertex number one

Given an instance of the preferential attachment graph $G_n=([n],E_n)$, we would like to find vertex 1, using only 'local' information about the graph; that is, by exploring the neighborhoods of small sets of vertices. Borgs et. al gave an an algorithm which runs in time $O(\log^4 n)$, which is local in the sense that at each step, it needs only to search the neighborhood of a set of vertices of size $O(\log^4 n)$. We give an algorithm to find vertex 1, which w.h.p. runs in time $O(\omega\log n)$ and which is local in the strongest sense of operating only on neighborhoods of single vertices. Here $\omega=\omega(n)$ is any function that goes to infinity with $n$.Comment: As accepted for AA

Rainbow vertex connection number and strong rainbow vertex connection number on slinky graph (SlnC4))

A graph is said rainbow connected if no path has more than one vertices of the same color inside. The minimum number of colors required to make a graph to be rainbow vertex-connected is called rainbow vertex connection-number and denoted by rvc(G) . Meanwhile, the minimum number of colors  required to make a graph to be strongly rainbow vertex-connected is called strong rainbow vertex connection-number and denoted by srvc(G). Suppose there is a simple, limited, and finite graph G. Thus, G=(V(G),E(G)) with the determination of k-coloring c:V(G)-{1,2,...,k} . The reaserch aims at determining rainbow vertex connection-number and strong rainbow vertex connection-number on slinky graphs (Sl_nC_4). Moreover, the research method applies a literature study with the following procedures; drawing slinky graphs (Sl_nC_4), looking for patterns of rainbow vertex connection-number, and strong rainbow vertex connection-number on slinky graphs (Sl_nC_4), then proving the theorems obtained from the previous pattern. It is obtained rvc(Sl_nC_4)=2n-1, srvc(Sl_2C_4)=4, and srvc(Sl_nC_4) = 3n-3 for n= 3.

Structure and enumeration of (3+1)-free posets

A poset is (3+1)-free if it does not contain the disjoint union of chains of length 3 and 1 as an induced subposet. These posets play a central role in the (3+1)-free conjecture of Stanley and Stembridge. Lewis and Zhang have enumerated (3+1)-free posets in the graded case by decomposing them into bipartite graphs, but until now the general enumeration problem has remained open. We give a finer decomposition into bipartite graphs which applies to all (3+1)-free posets and obtain generating functions which count (3+1)-free posets with labelled or unlabelled vertices. Using this decomposition, we obtain a decomposition of the automorphism group and asymptotics for the number of (3+1)-free posets.Comment: 28 pages, 5 figures. New version includes substantial changes to clarify the construction of skeleta and the enumeration. An extended abstract of this paper appears as arXiv:1212.535

The action of long strings in supersymmetric field theories

Long strings emerge in many Quantum Field Theories, for example as vortices in Abelian Higgs theories, or flux tubes in Yang-Mills theories. The actions of such objects can be expanded in the number of derivatives, around a long straight string solution. This corresponds to the expansion of energy levels in powers of 1/L, with L the length of the string. Doing so reveals that the first few terms in the expansions are universal, and only from a certain term do they become dependent on the originating field theory. Such classifications have been made before for bosonic strings. In this work we expand upon that and classify also strings with fermionic degrees of freedom, where the string breaks D=4 N=1 SUSY completely. An example is the confining string in N=1 SYM theory. We find a general method for generating supersymmetric action terms from their bosonic counterparts, as well as new fermionic terms which do not exist in the non-supersymmetric case. These terms lead to energy corrections at a lower order in 1/L than in the bosonic case