A new shortest proof of Kotzig's Theorem about graphs with unique perfect matching is presented in this paper. It is well known that Kotzig's theorem is a consequence of Yeo's Theorem about edge-colored graph without alternating cycle. We present a proof of Yeo's Theorem based on the same ideas as our proof of Kotzig's theorem.
This paper is the first from a series of papers that establish a generalization of the basilica decomposition for cardinality minimum joins in grafts. Joins in grafts are also known as $T$-joins in graphs, where $T$ is a given set of vertices, and minimum joins in grafts can be considered as a generalization of perfect matchings in graphs provided in terms of parity. The basilica decomposition is a canonical decomposition applicable to general graphs with perfect matchings, and the general Kotzig-Lovász decomposition is one of the three central concepts that compose this theory. The classical …
In matching theory, one of the most fundamental and classical branches of combinatorics, {\em canonical decompositions} of graphs are powerful and versatile tools that form the basis of this theory. However, the abilities of the known canonical decompositions, that is, the {\em Dulmage-Mendelsohn}, {\em Kotzig-Lovász}, and {\em Gallai-Edmonds} decompositions, are limited because they are only applicable to particular classes of graphs, such as bipartite graphs, or they are too sparse to provide sufficient information. To overcome these limitations, we introduce a new canonical decomposition th…
In this note, we give short inductive proofs of two known results on $k$-extendible graphs based on a property proved in [Qinglin Yu, A note on $n$-extendable graphs. Journal of Graph Theory, 16:349-353, 1992].
We give a simple short proof of Brooks' theorem using only induction and greedy coloring, while avoiding issues of graph connectivity. The argument generalizes easily to some extensions of Brooks' theorem, including its variants for list coloring, signed graphs coloring and correspondence coloring.
We provide a short and self-contained proof of the classical result of Kostochka and of Thomason, ensuring that every graph of average degree $d$ has a complete minor of order $d/\sqrt{\log d}$.
The $n$-th rencontres number with the parameter $r$ is the number of permutations having exactly $r$ fixed points. In particular, a derangement is a permutation without any fixed point. We presents a short combinatorial proof for a weighted sum derangement identities.
We prove several results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are quite short.
We give a short proof that every finite graph (or matroid) has a tree-decomposition that displays all maximal tangles. This theorem for graphs is a central result of the graph minors project of Robertson and Seymour and the extension to matroids is due to Geelen, Gerards and Whittle.
We prove a selection of results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are reasonably short.
For integers $n\geq k\geq 1$, the Kneser graph $K(n,k)$ is the graph with vertex set $V=[n]^{(k)}$ and edge set $E=\{\{x,y\} \in V^{(2)}: x\cap y=\emptyset\}$. Chen proved that for $n\geq 3k$, Kneser graphs are Hamiltonian and later improved this to $n\geq 2.62k+1$. Furthermore, Chen and Füredi gave a short proof that if $k | n$, Kneser graphs are Hamiltonian for $n\geq 3k$. In this note, we present a short proof that does not need the divisibility condition, i.e., we give a short proof that $K(n,k)$ is Hamiltonian for $n\geq 4k$.
We deduce Narayana's formula for the number of lattice paths that fit in a Young diagram as a direct consequence of the Gessel-Viennot theorem on non-intersecting lattice paths.
We provide a short proof of a conjecture of Davila and Kenter concerning a lower bound on the zero forcing number $Z(G)$ of a graph $G$. More specifically, we show that $Z(G)\geq (g-2)(δ-2)+2$ for every graph $G$ of girth $g$ at least $3$ and minimum degree $δ$ at least $2$.
"V - E + F = 2", the famous Euler's polyhedral formula, has a natural generalization to convex polytopes in every finite dimension, also known as the Euler-Poincaré Formula. We provide another short inductive proof of the general formula. Our proof is self-contained and it does not use shellability of polytopes.
We provide a short proof on the change-of-basis coefficients from the Specht basis to the Kazhdan-Lusztig basis, using Kazhdan-Lusztig theory for parabolic Hecke algebra.
We present a new proof of the well known formula for the rank of the inclusion matrix by constructing a $k\mathcal{S}_n$-module spanned by the columns of this matrix and calculating its dimension.
Here we present a short proof that the two-color van der Waerden number $w(3,k)$ is bounded from below by $(1-o(1))k^2$. Previous work has already shown that a superpolynomial lower bound holds for $w(3,k)$. However, we believe our result is still is of interest due to our techniques.
Battle, Harary, and Kodama (1962) and independently Tutte (1963) proved that the complete graph with nine vertices is not biplanar. Aiming towards simplicity and brevity, in this note we provide a short proof of this claim.
