An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

Ramsey-nice families of graphs

For a finite family $\mathcal{F}$ of fixed graphs let $R_k(\mathcal{F})$ be the smallest integer $n$ for which every $k$-coloring of the edges of the complete graph $K_n$ yields a monochromatic copy of some $F\in\mathcal{F}$. We say that $\mathcal{F}$ is $k$-nice if for every graph $G$ with $\chi(G)=R_k(\mathcal{F})$ and for every $k$-coloring of $E(G)$ there exists a monochromatic copy of some $F\in\mathcal{F}$. It is easy to see that if $\mathcal{F}$ contains no forest, then it is not $k$-nice for any $k$. It seems plausible to conjecture that a (weak) converse holds, namely, for any finite family of graphs $\mathcal{F}$ that contains at least one forest, and for all $k\geq k_0(\mathcal{F})$ (or at least for infinitely many values of $k$), $\mathcal{F}$ is $k$-nice. We prove several (modest) results in support of this conjecture, showing, in particular, that it holds for each of the three families consisting of two connected graphs with 3 edges each and observing that it holds for any family $\mathcal{F}$ containing a forest with at most 2 edges. We also study some related problems and disprove a conjecture by Aharoni, Charbit and Howard regarding the size of matchings in regular 3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Counting Hamilton decompositions of oriented graphs

A Hamilton cycle in a directed graph G is a cycle that passes through every vertex of G⁠. A Hamilton decomposition of G is a partition of its edge set into disjoint Hamilton cycles. In the late 60s, Kelly conjectured that every regular tournament has a Hamilton decomposition. This conjecture was recently settled for large tournaments by Kühn and Osthus [13], who proved more generally that every r-regular n-vertex oriented graph G (without antiparallel edges) with r=cn for some fixed c>3/8 has a Hamilton decomposition, provided n=n(c) is sufficiently large. In this article, we address the natural question of estimating the number of such decompositions of G and show that this number is n(1−o(1))cn^2⁠. In addition, we also obtain a new and much simpler proof for the approximate version of Kelly’s conjecture

Spartan Daily, March 27, 1987

Volume 88, Issue 42https://scholarworks.sjsu.edu/spartandaily/7568/thumbnail.jp

Spartan Daily, June 1, 1939

Volume 27, Issue 148https://scholarworks.sjsu.edu/spartandaily/2938/thumbnail.jp

Spartan Daily, June 1, 1939

Volume 27, Issue 148https://scholarworks.sjsu.edu/spartandaily/2938/thumbnail.jp

Spartan Daily, June 1, 1939

Volume 27, Issue 148https://scholarworks.sjsu.edu/spartandaily/2938/thumbnail.jp