| Abstract: | 假設F、G、H為三個圖，若H為G的一個生成子圖，且H中的每個分支都與F同構，則稱G有一個F-因子。令G和F為兩個圖，若G可分割成G_1,G_2,…,G_n，且每個G_i均為G的F-因子，則稱G有F-因子分解。在論文中，我們探討K_{m,n}的P_t-因子分解問題時，將t分為偶數和奇數來討論。首先，當t為偶數時，我們分別得到(1)若m為正整數，則K_{m,m}有P_2-因子分解。(2)當t為大於等於2的正整數時，若K_{m,n}有P_t-因子分解，則對於每一個正整數s，K_{ms,ns}有P_t-因子分解。(3)K_{m,n}有P_2k-因子分解的充分必要條件為m=n且m≡0(mod k(2k-1))。最後，當t為奇數時，我們分別獲得(1)若k為奇數，對於所有正整數s，則K_{ks,(k+1)s}有P_2k+1-因子分解。(2)若k為正整數，對於所有正整數s，則K_{2ks,2(k+1)s}有P_2k+1-因子分解。(3) K_{m,m}有P_2k+1-因子分解的充分必要條件為m≡0(mod 4k(2k+1))。
Suppose F,G and H are three graphs.If H is a spanning subgraph of G and each components of H is isomorphic to F,then G has F-factor.Let G and F are two graphs.If G can decompose G_1,G_2, … , G_n,and G_i is F-factor of G,then G has F-factorization.In this thesis,we discuss the problem about K_{m,n} has P_t-factorization.We will discuss as t is odd and even.First,we discuss as t is odd.We obtain three result:(1)If m is positive, then K_{m,m} has P_2-factorization.(2)Let t is positive integer and t >=2,if K_{m,n} has P_t-factorization,then K_{ms,ns} has P_t-factorization,for all s is positive integer.(3)K_{m,n} has P_2k-factorization if and only if m=n and m≡0(mod k(2k-1)).At last,we discuss as t is even.We get three result:(1)If k is odd and k is positive integer,for all s is positive integer,then K_{ks,(k+1)s} has P_2k+1-factorization.(2)If k is positive integer,for all s is positive integer, then K_{2ks,2(k+1)s} has P_2k+1-factorization.(3)K_{m,n} has P_2k+1-factorization if and only if m≡0(mod 4k(2k+1)). |
