Related papers: A different short proof of Brooks' theorem
L. Weinstein's brilliant short proof of de Branges's Theorem is made even shorter by using computer algebra.
K\"onig's edge coloring theorem says that a bipartite graph with maximal degree $n$ has an edge coloring with no more than $n$ colors. We explore the computability theory and Reverse Mathematics aspects of this theorem. Computable bipartite…
We work with simple graphs in ZF (Zermelo--Fraenkel set theory without the Axiom of Choice (AC)) and assume that the sets of colors can be either well-orderable or non-well-orderable to prove that the following statements are equivalent to…
The Lovasz Local Lemma is a seminal result in probabilistic combinatorics. It gives a sufficient condition on a probability space and a collection of events for the existence of an outcome that simultaneously avoids all of those events.…
Dvo\v{r}\'{a}k and Postle \cite{DP} introduced a \textit{DP-coloring} of a simple graph as a generalization of a list-coloring. They proved a Brooks' type theorem for a DP-coloring, and Bernshteyn, Kostochka and Pron \cite{BKP} extended it…
The Lov\'{a}sz Local Lemma is a central tool in probabilistic combinatorics, providing a sufficient condition under which a finite collection of undesirable events with limited dependencies can be simultaneously avoided with positive…
In [J. Combin. Theory Ser. B 70 (1997), 2-44] we gave a simplified proof of the Four-Color Theorem. The proof is computer-assisted in the sense that for two lemmas in the article we did not give proofs, and instead asserted that we have…
We prove a better coloring theorem for aleph_4 and even aleph_3. This has a general topology consequence.
We provide a simple and short proof of the Karush-Kuhn-Tucker theorem with finite number of equality and inequality constraints. The proof relies on an elementary linear algebra lemma and the local inverse theorem.
It was recently proved that every claw-free cubic graph admits a (1, 1, 2, 2)-packing coloring--that is, its vertex set can be partitioned into two 1-packings and two 2-packings. This result was established by Bre\v{s}ar, Kuenzel, and Rall…
We prove that the entropy compression method systematized by L. Esperet and A. Parreau can be applied to any problem formulated in the variable version of the Lovasz Local Lemma. As an application, we prove the existence of legitimate…
The constructive Lov\'{a}sz Local Lemma has become a central tool for designing efficient distributed algorithms. While it has been extensively studied in the classic LOCAL model that uses unlimited bandwidth, much less is known in the…
We prove the lemma of Knaster-Kuratowski-Mazurkiewicz as a consequence of the Lusternik-Schnirelman-Borsuk theorem.
We give a simple reformulation of the four color theorem as a problem on strings over a four letter alphabet.
Vizing's theorem states that every simple undirected graph can be edge-colored using fewer than $\Delta + 1$ colors, where $\Delta$ is the graph's maximum degree. The original proof was given through a polynomial-time algorithmic procedure…
P. Kirchberger proved that, for a finite subset $X$ of $\mathbb{R}^{d}$ such that each point in $X$ is painted with one of two colors, if every $d+2$ or fewer points in $X$ can be separated along the colors, then all the points in $X$ can…
A very short proof of the Fej\'er-Riesz lemma is presented in the matrix case
We give a simple proof of Strassen's theorem on stochastic dominance using linear programming duality, without requiring measure-theoretic arguments. The result extends to generalized inequalities using conic optimization duality and…
Boolos's proof of incompleteness is extended straightforwardly to yield simple ``diagonalization-free'' proofs of some classical limitative theorems of logic.
Kneser's 1955 conjecture -- proven by Lov\'asz in 1978 -- asserts that in any partition of the $k$-subsets of $\{1, 2, \dots, n\}$ into $n-2k-3$ parts, one part contains two disjoint sets. Schrijver showed that one can restrict to…