http://www.columbia.edu/~ww2040/GenFctORletters.pdf

Numerical inversion of probability generating functions

It is well known that Moore digraphs do not exist except for trivial cases (degree 1 or diameter 1), but there are digraphs of diameter two and arbitrary degree which miss the Moore bound by one. No examples of such digraphs of diameter at least three are known, although several necessary conditions for their existence have been obtained. In this paper, we prove that digraphs of degree three and diameter k ≥ 3 which miss the Moore bound by one do not exist. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 112–126, 2005

Complete characterization of almost Moore digraphs of degree three

We show that for any 2--coloring of the ${n \choose 2}$ segments determined by $n$ points in the plane, one of the color classes contains non-crossing cycles of lengths $3,4,\ldots,\lfloor\sqrt{n/2}\rfloor$. This result is tight up to a multiplicative constant. Under the same assumptions, we also prove that there is a non-crossing path of length $\Omega(n^{2/3})$, all of whose edges are of the same color. In the special case when the $n$ points are in convex position, we find longer monochromatic non-crossing paths, of length $\lceil\frac{n+1}{2}\rceil$. This bound cannot be improved. We also discuss some related problems and generalizations. In particular, we give sharp estimates for the largest number of disjoint monochromatic triangles that can always be selected from our segments.

Ramsey-type results for geometric graphs. II

The main treasure that Paul Erd˝ os has left us is his collection of problems, most of which are still open today. These problems are seeds that Paul sowed and watered by giving numerous talks at meetings big and small, near and far. In the past, his problems have spawned many areas in graph theory and beyond (e.g., in number theory, probability, geometry, algorithms and complexity theory). Solutions or partial solutions to Erd˝ os problems usually lead to further questions, often in new directions. These problems provide inspiration and serve as a common focus for all graph theorists. Through the problems, the legacy of Paul Erd˝ os continues (particularly if solving one of these problems results in creating three new problems, for example.) There is a huge literature of almost 1500 papers written by Erd˝ os and his (more than 460) collaborators. Paul wrote many problem papers, some of which appeared in various (really hard-to-find) proceedings. Here is an attempt to collect and organize these problems in the area of graph theory. The list here is by no means complete or exhaustive. Our goal is to state the problems, locate the sources, and provide the references related to these problems. We will include the earliest and latest known references without covering the entire history of the problems because of space limitations. (The most up-to-date list of Erd˝ os’ papers can be found in [65]; an electronic file is maintained by Jerry Grossman at grossman@oakland.edu.) There are many survey papers on the impact of Paul’s work, e.g., see those in the books: “A Tribute to Paul Erd˝ os” [84], “Combinatorics, Paul Erd˝ os is Eighty”, Volumes 1 and 2 [83], and “The Mathematics of Paul Erd˝ os”, Volumes I and II [81]. To honestly follow the unique style of Paul Erd˝ os, we will mention the fact that Erd˝ os often offered monetary awards for solutions to a number of his favorite problems. In November 1996, a committee of Erd˝ os’ friends decided no more such awards will be given in Erd˝ os’ name. However, the author, with the help of Ron Graham, will honor future claims on the problems in this paper, provided the requirements previously set by Paul are satisfied (e.g., proofs have been verified and published in recognized journals).

/pdf/open-problems-of-paul-erdos-in-graph-theory-4fshxfi7wp.pdf

Open problems of Paul Erdos in graph theory

Here we address the problem to partition edge colored hypergraphs by monochromatic paths and cycles generalizing a well-known similar problem for graphs. We show that $r$-colored $r$-uniform complete hypergraphs can be partitioned into monochromatic Berge-paths of distinct colors. Also, apart from $2k-5$ vertices, $2$-colored $k$-uniform hypergraphs can be partitioned into two monochromatic loose paths. In general, we prove that in any $r$-coloring of a $k$-uniform hypergraph there is a partition of the vertex set into monochromatic loose cycles such that their number depends only on $r$ and $k$.

/pdf/monochromatic-path-and-cycle-partitions-in-hypergraphs-4fzetq45cu.pdf

Monochromatic Path and Cycle Partitions in Hypergraphs

Vertex coverings by Monochromatic cycles and trees

Covering the Edges of a Connected Graph by Paths

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Vertex coverings by Monochromatic cycles and trees

Covering the Edges of a Connected Graph by Paths