On the Ramsey Numbers of Odd-Linked Double Stars

TL;DR

This paper investigates the Ramsey numbers of odd-linked double star graphs, establishing bounds and exact values under specific conditions, thereby advancing understanding of their monochromatic subgraph properties in 2-colorings.

Contribution

It provides new bounds and exact values for the Ramsey numbers of linked double stars when the linking path length is odd, filling gaps in existing combinatorial graph theory knowledge.

Findings

Established bounds for $r(S_c(n,m))$ with odd $c$.

Determined exact Ramsey numbers when $n \

n \

Abstract

The linked double star $S_c(n,m)$, where $n \geq m \geq 0$, is the graph consisting of the union of two stars $K_{1,n}$ and $K_{1,m}$ with a path on $c$ vertices joining the centers. Its ramsey number $r(S_c(n,m))$ is the smallest integer $r$ such that every $2$-coloring of the edges of a $K_r$ admits a monochromatic $S_c(n,m)$. In this paper, we study the ramsey numbers of linked double stars when $c$ is odd. In particular, we establish bounds on the value of $r(S_c(n,m))$ and determine the exact value of $r(S_c(n,m))$ if $n \geq c$, or if $n \leq \lfloor \frac{c}{2} \rfloor - 2$ and $m = 2$.