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Ramsey Numbers for Trees II
Czechoslovak Mathematical Journal ( IF 0.4 ) Pub Date : 2021-03-05 , DOI: 10.21136/cmj.2021.0328-19
Zhi-Hong Sun

Let r(G1, G2) be the Ramsey number of the two graphs G1 and G2. For n1n2 ⩾ 1 let S(n1, n2) be the double star given by \(V\left( {S\left( {{n_1},{n_2}} \right)} \right) = \left\{ {{v_0},{v_1}, \ldots ,{v_{{n_1}}},{w_0},{w_1}, \ldots ,{w_{{n_2}}}} \right\}\) and \(E\left( {S\left( {{n_1},{n_2}} \right)} \right) = \left\{ {{v_0}{v_1}, \ldots ,{v_0}{v_{{n_1}}},{v_0}{w_0},{w_0}{w_1}, \ldots ,{w_0}{w_{{n_2}}}} \right\}\). We determine r(K1,m−1, S(n1, n2)) under certain conditions. For n ⩾ 6 let \(T_n^3 = S(n - 5,3),T_n^{\prime\prime} = (V,{E_2})\) and \(T_n^{\prime\prime\prime} = (V,{E_3})\), where \(V = \left\{ {{v_0},{v_1}, \ldots ,{v_{n - 1}}} \right\},\;{E_2} = \left\{ {{v_0}{v_1}, \ldots ,{v_0}{v_{n - 4}},{v_1}{v_{n - 3}},{v_1}{v_{n - 2}},{v_2}{v_{n - 1}}} \right\}\) and \({E_3} = \left\{ {{v_0}{v_1}, \ldots ,{v_0}{v_{n - 4}},{v_1}{v_{n - 3}},{v_2}{v_{n - 2}},{v_3}{v_{n - 1}}} \right\}\). We also obtain explicit formulas for \(r({K_{1,m - 1}},{T_n}),\;r(T_m^\prime,{T_n})\;(n \geqslant m + 3),\;r({T_n},{T_n}),\;r(T_n^\prime,{T_n})\) and r(Pn, Tn), where \({T_n} \in \left\{ {T_n^{\prime\prime},T_n^{\prime\prime\prime},T_n^3} \right\}\), Pn is the path on n vertices and \(T_n^\prime\) is the unique tree with n vertices and maximal degree n − 2.



中文翻译:

树的拉姆齐编号II

rG 1G 2)为两个图G 1G 2的Ramsey数。对于Ñ 1Ñ 2 ⩾1让小号Ñ 1Ñ 2)是由下式给出的双星\(V \左({S \左({{N_1},{N_2}} \右)} \右) = \ left \ {{{v_0},{v_1},\ ldots,{v _ {{n_1}}},{w_0},{w_1},\ ldots,{w _ {{n_2}}}} \ right \} \)\(E \ left({S \ left({{n_1},{n_2}} \ right)} \ right)= \ left \ {{{v_0} {v_1},\ ldots,{v_0} { v _ {{n_1}},{v_0} {w_0},{w_0} {w_1},\ ldots,{w_0} {w _ {{n_2}}}} \ right \} \)。我们在特定条件下确定rK 1,m -1Sn 1n 2))。对于Ñ ⩾6令\(T_n ^ 3 = S(N - 5,3),T_n ^ {\素\素} =(V,{E_2})\)\(T_n ^ {\素\素\素} =(V,{E_3})\),其中\(V = \ left \ {{{v_0},{v_1},\ ldots,{v_ {n-1}}}} \ right \},\; { E_2} = \ left \ {{{v_0} {v_1},\ ldots,{v_0} {v_ {n-4}},{v_1} {v_ {n-3}},{v_1} {v_ {n- 2}},{v_2} {v_ {n-1}}} \ right \} \)\({E_3} = \ left \ {{{v_0} {v_1},\ ldots,{v_0} {v_ { n-4}},{v_1} {v_ {n-3}},{v_2} {v_ {n-2}},{v_3} {v_ {n-1}}} \ right \} \)。我们还获得了以下公式的显式\(r({K_ {1,m-1}},{T_n}),\; r(T_m ^ \ prime,{T_n})\;(n \ geqslant m + 3),\; r({T_n },{T_n}),\; r(T_n ^ \ prime,{T_n})\)rP nT n),其中\({T_n} \ in \ left \ {{T_n ^ {\ prime \ prime},T_n ^ {\ prime \ prime \ prime},T_n ^ 3} \ right \} \)P nn个顶点上的路径,\(T_n ^ \ prime \)是具有n个顶点的唯一树和最大度数n − 2

更新日期:2021-04-11
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