Related papers: Yet another short proof of the 4-colour theorem
This paper has been withdrawn by the author(s), due an error in the proof.
A short proof is given that the graphs with proper interval representations are the same as the graphs with unit interval representations.
It is shown that contrary to claims of Ref. [1] the formulated in the proper physical variables Low theorem [2] for soft photon emission does not require any modification. We also reject the criticism in Ref. [1] of the papers [3,4]. At the…
We present an alternative proof of Perron's theorem, which is probabilistic in nature. It rests on the representation of the Perron eigenvector as a functional of the trajectory of an auxiliary Markov chain.
We consider extensions of Brooks' classic theorem on vertex coloring where some colors cannot be used on certain vertices. In particular we prove that if $G$ is a connected graph with maximum degree $\Delta(G) \geq 4$ that is not a complete…
Popularized explanations of the color of the sea often prove to be incomplete or oversimplified, and hence inadequate to become acquainted with the phenomenon in its whole. In this paper, after a historical review of the investigations on…
In this small note we ask several questions which are relevant to the construction of the self-consistent neutrino theory of light. The previous confusions in such attempts are explained in the more detailed publication.
In this survey essay, I explore the application of the discharging method in graph theory, including the selection of charging rules and discharging rules, and the general characteristics of the discharging method. As examples, I will prove…
This paper has been withdrawn by the author due to an error in the main proof (thanks to Carlos D'Andrea)
For a plane near-triangulation $G$ with the outer face bounded by a cycle $C$, let $n^\star_G$ denote the function that to each $4$-coloring $\psi$ of $C$ assigns the number of ways $\psi$ extends to a $4$-coloring of $G$. The block-count…
"Systems that Explain Themselves" appears a provocative wording, in particular in the context of mathematics education -- it is as provocative as the idea of building educational software upon technology from computer theorem proving. In…
This paper studies the proof of Collatz conjecture for some set of sequence of odd numbers with infinite number of elements. These set generalized to the set which contains all positive odd integers. This extension assumed to be the proof…
The definition of the weighted version of difficulty in "Termination of 4-fold canonical flips" contains some errors. In this paper, we describe these errors and how to correct them. Anyway, any sequence of 4-fold canonical flips…
Defective coloring is a variant of traditional vertex-coloring, according to which adjacent vertices are allowed to have the same color, as long as the monochromatic components induced by the corresponding edges have a certain structure.…
Fix a palette $\mathcal K$ of $\Delta+1$ colours, a graph with maximum degree $\Delta$, and a subset $M$ of the edge set with minimum distance between edges at least $9$. If the edges of $M$ are arbitrarily precoloured from $\mathcal K$,…
We prove upper and lower bounds on the effective content and logical strength for a variety of natural restrictions of Hindman's Finite Sums Theorem. For example, we show that Hindman's Theorem for sums of length at most 2 and 4 colors…
Chromatic quantum contextuality is a criterion of quantum nonclassicality based on (hyper)graph coloring constraints. If a quantum hypergraph requires more colors than the number of outcomes per maximal observable (context), it lacks a…
The recently developed proof of Fermat's Last Theorem is very lengthy and difficult, so much so as to be beyond all but a small body of specialists. While certainly of value in the developments that resulted, that proof could not be, nor…
This is an exposition of Gauss's proof of Descartes's rule of signs.
Proofs that a smooth morphism is flat available in the literature are long and difficult. We give a short proof of this fact.