Combinatorics and Graph Theory

2603 Submissions

[3] viXra:2603.0086 [pdf] submitted on 2026-03-17 00:22:13

Lonely Runner Conjecture Proof

Authors: Bhaskar Kumar
Comments: 2 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

The Lonely Runner Conjecture, independently proposed by Wills (1967) and Cusick (1973), asserts that for any n runners on a unit circular track moving at distinct integer speeds, there exists a time at which each runner is at distance at least 1/n from a fixed reference runner. We provide a complete proofby reformulating the problem in terms of simultaneous residue conditions and proving existence via an inductive construction using the least common multiple and arithmetic properties of congruences. The proof is elementary, relying only on the Chinese Remainder Theorem and basic number-theoretic tools.
Category: Combinatorics and Graph Theory

[2] viXra:2603.0059 [pdf] submitted on 2026-03-11 20:25:37

Asymptotic Behavior of the Erdos Consecutive Sum Sequence A003462

Authors: Jehan Singh
Comments: 4 Pages. CC BY 4.0 license (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)

We study the self-generating consecutive-sum greedy sequence (an)n ≥ 1, defined by a1 = 1, a2 = 2, and for k ≥ 3, ak is the least integer greater than ak−1 expressible as a sum of at least two consecutive earlier terms. We prove that the auxiliary sequence bn := an − n is nondecreasing and unbounded, showing that infinitely many positive integers are omitted. We also provide an explicit Binet-type formula for an, enabling exact computation of terms, and discuss implications for the sequence’s asymptotic growth.
Category: Combinatorics and Graph Theory

[1] viXra:2603.0043 [pdf] submitted on 2026-03-07 09:52:56

Dimensions and Coloring of a Graph

Authors: Volker Thürey
Comments: 7 Pages.

In the first part, we introduce several notions of graph dimensions. These concepts are inspired by the classical idea of a `dimension' which was introduced around 60 years ago. We provide simple examples to illustrate them. In the second part, we define new types of graph colorings, derived from the Chromatic Number and the Chromatic Index. We apply coloring to both the vertices and the the edges. Finally, we generalize the concept of the `Chromatic Number of the Plane'.
Category: Combinatorics and Graph Theory