Random multi-index matching problems

The multi-index matching problem (MIMP) generalizes the well known matching problem by going from pairs to d-uplets. We use the cavity method from statistical physics to analyze its properties when the costs of the d-uplets are random. At low temperatures we find for d>2 a frozen glassy phase with vanishing entropy. We also investigate some properties of small samples by enumerating the lowest cost matchings to compare with our theoretical predictions.Comment: 22 pages, 16 figure

Proximity and Remoteness in Directed and Undirected Graphs

Let $D$ be a strongly connected digraph. The average distance $\bar{\sigma}(v)$ of a vertex $v$ of $D$ is the arithmetic mean of the distances from $v$ to all other vertices of $D$. The remoteness $\rho(D)$ and proximity $\pi(D)$ of $D$ are the maximum and the minimum of the average distances of the vertices of $D$, respectively. We obtain sharp upper and lower bounds on $\pi(D)$ and $\rho(D)$ as a function of the order $n$ of $D$ and describe the extreme digraphs for all the bounds. We also obtain such bounds for strong tournaments. We show that for a strong tournament $T$, we have $\pi(T)=\rho(T)$ if and only if $T$ is regular. Due to this result, one may conjecture that every strong digraph $D$ with $\pi(D)=\rho(D)$ is regular. We present an infinite family of non-regular strong digraphs $D$ such that $\pi(D)=\rho(D).$ We describe such a family for undirected graphs as well

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

On the forced unilateral orientation number of a graph

AbstractA graph has a unilateral orientation if its edges can be oriented such that for every two vertices u and v there exists either a path from u to v or a path from v to u. If G is a graph with a unilateral orientation, then the forced unilateral orientation number of G is defined to be the minimum cardinality of a subset of the set of edges for which there is an assignment of directions that has a unique extension to a unilateral orientation of G. This paper gives a general lower bound for the forced unilateral orientation number and shows that the unilateral orientation number of a graph of size m, order n, and having edge connectivity 1 is equal to m − n + 2. A few other related problems are discussed

Quick Trips: On the Oriented Diameter of Graphs

In this dissertation, I will discuss two results on the oriented diameter of graphs with certain properties. In the first problem, I studied the oriented diameter of a graph G. Erdos et al. in 1989 showed that for any graph with |V | = n and δ(G) = δ the maximum the diameter could possibly be was 3 n/ δ+1. I considered whether there exists an orientation on a given graph with |G| = n and δ(G) = δ that has a small diameter. Bau and Dankelmann (2015) showed that there is an orientation of diameter 11 n/ δ+1 + O(1), and showed that there is a graph which the best orientation admitted is 3 n/ δ+1 + O(1). It was left as an open question whether the factor of 11 in the first result could be reduced to 3. The result above was improved to 7 n / δ+1 +O(1) by Surmacs (2017) and I will present a proof of a further improvement of this bound to 5 n/δ−1 + O(1). It remains open whether 3 is the best answer. In the second problem, I studied the oriented diameter of the complete graph Kn with some edges removed. We will show that given Kn with n \u3e= 5 and any collection of edges Ev, with |Ev| = n − 5, that there is an orientation of this graph with diameter 2. It remains a question how many edges we can remove to guarantee larger diameters

Diameter of orientations of graphs with given order and number of blocks

A strong orientation of a graph $G$ is an assignment of a direction to each edge such that $G$ is strongly connected. The oriented diameter of $G$ is the smallest diameter among all strong orientations of $G$. A block of $G$ is a maximal connected subgraph of $G$ that has no cut vertex. A block graph is a graph in which every block is a clique. We show that every bridgeless graph of order $n$ containing $p$ blocks has an oriented diameter of at most $n-\lfloor \frac{p}{2} \rfloor$. This bound is sharp for all $n$ and $p$ with $p \geq 2$. As a corollary, we obtain a sharp upper bound on the oriented diameter in terms of order and number of cut vertices. We also show that the oriented diameter of a bridgeless block graph of order $n$ is bounded above by $\lfloor \frac{3n}{4} \rfloor$ if $n$ is even and $\lfloor \frac{3(n+1)}{4} \rfloor$ if $n$ is odd.Comment: 15 pages, 2 figure