Abstract
The weakened Ramsey number $r^{s,t}(G)$ is defined to be the least $p\in \mathbb{N}$ such that every $t$-coloring of the edges of the complete graph $K_p$ contains a subgraph isomorphic to $G$ that is spanned by edges that use at most $s$ colors ($1\le s\le t-1$). The star-critical weakened Ramsey number $r^{s,t}_*(G)$ then determines the minimum number of edges that must join a vertex to $K_{r^{s,t}(G)-1}$ in order for this Ramsey property to hold. We begin by showing that $r_*^{s,t}(K_n)=r^{s,t}(K_n)-1$ for all $n\in \mathbb{N}$. Then, building off of Khamseh and Omidi's recent determination of $r^{s,t}(K_{1,n})$ when $s=t-1$ and $s=t-2$, we focus on the evaluation of $r_*^{t-1,t}(K_{1,n})$ and $r_*^{t-2,t}(K_{1,n})$. Additionally, we determine $r^{2,t}(K_{1,3})$ and $r_*^{2,t}(K_{1,3})$ when $t\ge 3$.
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.
Recommended Citation
Budden, Mark R.; Moun, Monu; and Jakhar, Jagjeet
(2025)
"Star-Critical Weakened Ramsey Numbers,"
Theory and Applications of Graphs: Vol. 12:
Iss.
2, Article 4.
DOI: 10.20429/tag.2025.120204
Available at:
https://digitalcommons.georgiasouthern.edu/tag/vol12/iss2/4
Supplemental Dois