Edge-coloring via fixable subgraphs

Many graph coloring proofs proceed by showing that a minimal counterexample to the theorem being proved cannot contain certain configurations, and then showing that each graph under consideration contains at least one such configuration; these configurations are called \emph{reducible} for that theorem. (A \emph{configuration} is a subgraph $H$, along with specified degrees $d_G(v)$ in the original graph $G$ for each vertex of $H$.) We give a general framework for showing that configurations are reducible for edge-coloring. A particular form of reducibility, called \emph{fixability}, can be considered without reference to a containing graph. This has two key benefits: (i) we can now formulate necessary conditions for fixability, and (ii) the problem of fixability is easy for a computer to solve. The necessary condition of \emph{superabundance} is sufficient for multistars and we conjecture that it is sufficient for trees as well, which would generalize the powerful technique of Tashkinov trees. Via computer, we can generate thousands of reducible configurations, but we have short proofs for only a small fraction of these. The computer can write \LaTeX\ code for its proofs, but they are only marginally enlightening and can run thousands of pages long. We give examples of how to use some of these reducible configurations to prove conjectures on edge-coloring for small maximum degree. Our aims in writing this paper are (i) to provide a common context for a variety of reducible configurations for edge-coloring and (ii) to spur development of methods for humans to understand what the computer already knows.Comment: 18 pages, 8 figures; 12-page appendix with 39 figure

On the classification problem for split graphs

Abstract The Classification Problem is the problem of deciding whether a simple graph has chromatic index equal to Δ or Δ+1. In the first case, the graphs are called Class 1, otherwise, they are Class 2. A split graph is a graph whose vertex set admits a partition into a stable set and a clique. Split graphs are a subclass of chordal graphs. Figueiredo at al. (J. Combin. Math. Combin. Comput. 32:79–91, 2000) state that a chordal graph is Class 2 if and only if it is neighborhood-overfull. In this paper, we give a characterization of neighborhood-overfull split graphs and we show that the above conjecture is true for some split graphs

Hamilton cycles in graphs and hypergraphs: an extremal perspective

As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasi-randomness. These concepts and other recent techniques have led to the solution of several long-standing problems in the area. New aspects have also emerged, such as resilience, robustness and the study of Hamilton cycles in hypergraphs. We survey these developments and highlight open problems, with an emphasis on extremal and probabilistic approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page limits, this final version is slightly shorter than the previous arxiv versio

Graph edge coloring and a new approach to the overfull conjecture

The graph edge coloring problem is to color the edges of a graph such that adjacent edges receives different colors. Let $G$ be a simple graph with maximum degree $\Delta$. The minimum number of colors needed for such a coloring of $G$ is called the chromatic index of $G$, written $\chi\u27(G)$. We say $G$ is of class one if $\chi\u27(G)=\Delta$, otherwise it is of class 2. A majority of edge coloring papers is devoted to the Classification Problem for simple graphs. A graph $G$ is said to be \emph{overfull} if $|E(G)|\u3e\Delta \lfloor |V(G)|/2\rfloor$. Hilton in 1985 conjectured that every graph $G$ of class two with $\Delta(G)\u3e\frac{|V(G)|}{3}$ contains an overfull subgraph $H$ with $\Delta(H)=\Delta(G)$. In this thesis, I will introduce some of my researches toward the Classification Problem of simple graphs, and a new approach to the overfull conjecture together with some new techniques and ideas

Short proofs of three results about intersecting systems

In this note, we give short proofs of three theorems about intersection problems. The first one is a determination of the maximum size of a nontrivial $k$-uniform, $d$-wise intersecting family for $n\ge \left(1+\frac{d}{2}\right)(k-d+2)$, which improves upon a recent result of O'Neill and Verstra\"{e}te. Our proof also extends to $d$-wise, $t$-intersecting families, and from this result we obtain a version of the Erd\H{o}s-Ko-Rado theorem for $d$-wise, $t$-intersecting families. The second result partially proves a conjecture of Frankl and Tokushige about $k$-uniform families with restricted pairwise intersection sizes. The third result concerns graph intersections. Answering a question of Ellis, we construct $K_{s, t}$-intersecting families of graphs which have size larger than the Erd\H{o}s-Ko-Rado-type construction whenever $t$ is sufficiently large in terms of $s$.Comment: 12 pages; we added a new result, Theorem 1