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S={1,2,…,m}为 m 元集,(?)(S)为 S 的子集全体.若(?)(?)(?)(S),记(?)={X|X∈(?),|X|=i}.设(?)(S)为 Sperner 系,即任意的 X_i、X_j,∈(?),X_i(?)X_j、若|(?)_i|=p_i,称{p_0,p_1,…,p(?)}为(?)的 Sperner 参数、1928年,Sperner 证明了 相似文献
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设k-i为正整数,i=1,2,…,n,直积S=Ⅰ_(R_1)×Ⅰ_(R_2)×…×Ⅰ_(k_n)={x_1,x_2,…,x_n,0≤x_i≤k_i}叫做链积,对任意的在偏序“<”下为有限偏序集。r(x)=sum from i(x_i)原S的秩函数,叫做S的Whitney数记k=sum from i=1 to n(k_i,k_1=k_2=k_n=1)时,S即为布尔代数B_n。 设为S中的反链,{P_i,i=0,1,…,n}叫做反链的参数,若成立 相似文献
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In a previous paper by the author joint with Baogang XU published in Discrete Math in 2018, we show that every non-planar toroidal graph can be edge partitioned into a planar graph and an outerplanar graph. This edge partition then implies some results in thickness and outerthickness of toroidal graphs. In particular, if each planar graph has outerthickness at most $2$ (conjectured by Chartrand, Geller and Hedetniemi in 1971 and the confirmation of the conjecture was announced by Gon\c{c}alves in 2005), then the outerthickness of toroidal graphs is at most 3 which is the best possible due to $K_7$. In this paper we continue to study the edge partition for projective planar graphs and Klein bottle embeddable graphs. We show that (1) every non-planar but projective planar graph can be edge partitioned into a planar graph and a union of caterpillar trees; and (2) every non-planar Klein bottle embeddable graph can be edge partitioned into a planar graph and a subgraph of two vertex amalgamation of a caterpillar tree with a cycle with pendant edges. As consequences, the thinkness of projective planar graphs and Klein bottle embeddabe graphs are at most $2$, which are the best possible, and the outerthickness of these graphs are at most $3$. 相似文献
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