Multicolor Ramsey Number of Stars Versus a Path 
Received:September 09, 2023 Revised:January 06, 2024 
Key Words:
Ramsey number star path

Fund Project:Supported by the National Natural Science Foundation of China (Grant No.12071453), the National Key R and D Program of China (Grant No.2020YFA0713100) and the Innovation Program for Quantum Science and Technology (Grant No.2021ZD0302902). 
Author Name  Affiliation  Xuejun ZHANG  School of Data Science, University of Science and Technology of China, Anhui 230026, P. R. China  Xinmin HOU  School of Data Science, University of Science and Technology of China, Anhui 230026, P. R. China School of Mathematical Sciences, University of Science and Technology of China, Anhui 230026, P. R. China CAS Key Laboratory of Wu WenTsun Mathematics, University of Science and Technology of China, Anhui 230026, P. R. China Hefei National Laboratory, University of Science and Technology of China, Anhui 230026, P. R. China 

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Abstract: 
For given simple graphs $H_1,H_2,\ldots,H_c$, the multicolor Ramsey number $R(H_1,H_2,\ldots,$ $H_c)$ is defined as the smallest positive integer $n$ such that for an arbitrary edgedecomposition $\{G_i\}^c_{i=1}$ of the complete graph $K_n$, at least one $G_i$ has a subgraph isomorphic to $H_i$. Let $m,n_1,n_2,\ldots,n_c$ be positive integers and $\Sigma=\sum_{i=1}^{c}(n_i1)$. Some bounds and exact values of $R(K_{1,n_1},\ldots,K_{1,n_c},P_m)$ have been obtained in literature. Wang (Graphs Combin., 2020) conjectured that if $\Sigma\not\equiv 0\pmod{m1}$ and $\Sigma+1\ge (m3)^2$, then $R(K_{1,n_1},\ldots, K_{1,n_c}, P_m)=\Sigma+m1$. In this note, we give a new lower bound and some exact values of $R(K_{1,n_1},\ldots,K_{1,n_c},P_m)$ provided $m\leq\Sigma$, $\Sigma\equiv k\pmod{m1}$, and $2\leq k \leq m2$. These results partially confirm Wang's conjecture. 
Citation: 
DOI:10.3770/j.issn:20952651.2024.04.004 
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