Related papers: A graph theoretic proof of Cramer's rule
Working in any model theoretic structure, we single out a class of definable bipartite graphs that admit definable, close to perfect matchings. We use this result to prove a strengthening of Tarski's theorem for the definable setting.
Consider a weighted directed acyclic graph $G$ having an upward planar drawing. We give a formula for the total weight of the families of non-intersecting paths on $G$ with any given starting and ending points. While the…
Roussel and Rubio proved a lemma which is essential in the proof of the Strong Perfect Graph Theorem. We give a new short proof of the main case of this lemma. In this note, we also give a short proof of Hayward's decomposition theorem for…
In the current note, we present a new, short proof of the famous AM-GM-HM inequality using only induction and basic calculus.
Elementary proofs of Sylvester's, Wolstenholme's, Morley's and Lehmer's congruence theorems
We give a new proof of the theorem of Kronecker-Weber based on Kummer theory and Stickelberger's theorem.
This note shows how Condition 1 in Okumura (2025) can be tested efficiently using a standard graph-theoretic algorithm. It also describes an efficient implementation of Okumura's mechanism.
We find rational expressions for all minors of the weighted path matrix of a directed graph, generalizing the classical Lindstrom/Gessel-Viennot result for acyclic directed graphs. The formulas are given in terms of certain flows in the…
We give an elementary combinatorial proof of Bass's determinant formula for the zeta function of a finite regular graph. This is done by expressing the number of non-backtracking cycles of a given length in terms of Chebychev polynomials in…
The purpose of this note is to rephrase Speyer's elegant topological proof for Kasteleyn's Theorem in a simple graph theoretical manner.
This note is devoted to a combinatorial proof of a Schmidt type theorem due to Andrews and Paule. A four-variable refinement of Andrews and Paule's theorem is also obtained based on this combinatorial construction.
Let $G$ be a graph, and let $\lambda(G)$ denote the smallest eigenvalue of $G$. First, we provide an upper bound for $\lambda(G)$ based on induced bipartite subgraphs of $G$. Consequently, we extract two other upper bounds, one relying on…
In this note, we combine ideas of several previous proofs in order to obtain a quite short proof of Gr\"otzsch theorem.
In this article we are introducing combinatorial spectra of graphs, this is a generalization of $H$-Hamiltonian spectra. The main motivation was to made from $H$-Hamiltonian spectra an operation and develop some algebra in this field. An…
In this paper, we will consider the graph w*-probability theory.
The Union Closed Sets Conjecture is one of the most renowned problems in combinatorics. Its appeal lies in the simplicity of its statement contrasted with the potential complexity of its resolution. The conjecture posits that, in any union…
We give a combinatorial upper bound for the gonality of a curve that is defined by a bivariate Laurent polynomial with given Newton polygon. We conjecture that this bound is generically attained, and provide proofs in a considerable number…
We provide a new proof of a theorem of Hell and Ne\v{s}et\v{r}il [J. Comb. Theory B, 48(1):92-110, 1990] using tools from topological combinatorics based on ideas of Lov\'asz [J. Comb. Theory, Ser. A, 25(3):319-324, 1978]. The…
A new kind of diagrams is presented, showing the causal structure of bimetric interactions.
In this note I give an information-theoretic proof of the Bonami-Beckner-Gross hypercontractive inequality.