TY - JOUR
T1 - On the 3-Color Ramsey Numbers R(C4, C4, Wn)
AU - Zhang, Xuemei
AU - Chen, Yaojun
AU - Cheng, T. C.Edwin
N1 - Funding Information:
Zhang was supported by NSFC under grant number 11801520 and ZJNSF under grant number LY18A010014, and Chen was supported by NSFC under Grant numbers 11671198, 11871270 and 12161141003.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Japan KK, part of Springer Nature.
PY - 2022/6
Y1 - 2022/6
N2 - For given graphs G1, G2, ⋯ , Gk, k≥ 2 , the k-color Ramsey number, denoted by R(G1, G2, … , Gk) , is the smallest integer N such that if we arbitrarily color the edges of a complete graph of order N with k colors, then it always contains a monochromatic copy of Gi in color i, for some 1 ≤ i≤ k. Let Cm be a cycle of length m and Wn a wheel of order n+ 1. In this paper, we show that R(C4,C4,Wn)≤n+⌈4n+5⌉+3 for n= 42 , 48 , 49 , 50 , 51 , 52 or n≥ 56. Furthermore, we prove that R(C4,C4,Wℓ2-ℓ)≤ℓ2+ℓ+2 for ℓ≥ 9 , and if ℓ is a prime power, then the equality holds.
AB - For given graphs G1, G2, ⋯ , Gk, k≥ 2 , the k-color Ramsey number, denoted by R(G1, G2, … , Gk) , is the smallest integer N such that if we arbitrarily color the edges of a complete graph of order N with k colors, then it always contains a monochromatic copy of Gi in color i, for some 1 ≤ i≤ k. Let Cm be a cycle of length m and Wn a wheel of order n+ 1. In this paper, we show that R(C4,C4,Wn)≤n+⌈4n+5⌉+3 for n= 42 , 48 , 49 , 50 , 51 , 52 or n≥ 56. Furthermore, we prove that R(C4,C4,Wℓ2-ℓ)≤ℓ2+ℓ+2 for ℓ≥ 9 , and if ℓ is a prime power, then the equality holds.
KW - Multicolor Ramsey number
KW - Quadrilateral
KW - Wheel
UR - http://www.scopus.com/inward/record.url?scp=85130774606&partnerID=8YFLogxK
U2 - 10.1007/s00373-022-02505-y
DO - 10.1007/s00373-022-02505-y
M3 - Journal article
AN - SCOPUS:85130774606
SN - 0911-0119
VL - 38
JO - Graphs and Combinatorics
JF - Graphs and Combinatorics
IS - 3
M1 - 103
ER -