非連通圖C4m-1∪C12m-8∪G的優(yōu)美標(biāo)號

2014-03-25 06:06:22吳躍生

沈陽大學(xué)學(xué)報(自然科學(xué)版) 2014年4期

關(guān)鍵詞：吉首標(biāo)號正整數(shù)

吳躍生

(華東交通大學(xué) 理學(xué)院, 江西南昌 330013)

1 主要概念

本文所討論的圖均為無向簡單圖,V(G)和E(G)分別表示圖G的頂點集和邊集,記號[m,n]表示整數(shù)集合{m,m+1,…,n},其中,m和n均為非負整數(shù),且滿足0≤m

圖的優(yōu)美標(biāo)號問題是組合數(shù)學(xué)中一個熱門課題[1-11].

本文討論非連通圖C4m-1∪C12m-8∪G的優(yōu)美性.

定義2[1]設(shè)f為G的一個優(yōu)美標(biāo)號,如果存在一個正整數(shù)k,使得對于任意的uv∈E(G),有

f(u)>k≥f(v) 或f(u)≤k

成立,則稱f為G的平衡標(biāo)號(或稱G有平衡標(biāo)號f),且稱k為f的特征.圖G稱為平衡二分圖(balanced bipartite graph).

顯然,若f為G的平衡標(biāo)號,則k是邊導(dǎo)出標(biāo)號為1的邊的兩個端點中標(biāo)號較小的頂點的標(biāo)號.

定義3[1]在平衡二分圖G中,設(shè)其優(yōu)美標(biāo)號θ的特征為k,并且θ(u0)=k,θ(v0)=k+1,則稱u0為G的二分點,v0為G的對偶二分點.

定義5[3-4]V(G)= {u1,u2,…,un}的每個頂點ui都粘接了ri條懸掛邊(ri為自然數(shù),i=1,2,…,n)所得到的圖,稱為圖G的(r1,r2,…,rn)-冠,簡記為G(r1,r2,…,rn). 特別地,當(dāng)r1=r2=… =rn=r時,稱為圖G的r-冠.圖G的0-冠就是圖G.

2 主要結(jié)果及其證明

定理對于任意正整數(shù)m,如果圖G是特征為k且缺k+6m-3標(biāo)號值的交錯圖(6m-3≤k+6m-3≤|E(G)|),那么非連通圖C4m-1∪C12m-8∪G存在缺標(biāo)號值k+1的優(yōu)美標(biāo)號.

θ(x2i)=6m+i+k-2 (i=1,2,…,2m-1);

θ(x2i-1)=10m-i+k-3 (i=1,2,…,m);

θ(x2m+1)=3m+k;

θ(x2i-1)=10m-i+k-2

(i=m+2,m+3,…,2m);

θ(y2i-1)=16m-8-i+k(i=1,2,…,6m-5);

θ(y12m-9)=k+22m-12;

下面證明θ是非連通圖C4m-1∪C12m-8∪G的優(yōu)美標(biāo)號.

(1)θ:X→[0,k]是單射(或雙射);

θ:Y→[k+16m-8,q+16m-9]-{k+22m-12}是單射(或雙射);

θ:V(C4m-1)→[k+6m-1,10m-4+k]∪{3m+k}是單射(或雙射);

θ:V(C12m-8)→[k+2,6m-2+k]∪[10m-3+k,16m-9+k]∪{22m-12+k}-{3m+k}是單射(或雙射);

θ:V(C4m-1∪C12m-8∪G)→[0,q+16m-9]-{k+1}是單射.

θ′(x2m+2x2m+1)=4m-1;

θ′(x2mx2m+1)=4m-2,θ′(x4m-1x1)=2m-2;

θ′:E(C4m-1)→[1,4m-1]是雙射;

θ′(y12m-9y12m-8)=16m-10;

θ′(y12m-10y12m-9)=16m-9,θ′(y12m-8y1)=10m-7;

θ′:E(C12m-8)→[4m,12m-9]是雙射;

θ′:E(G)→[16m-8,q+16m-9]是雙射;

由(1)和(2)可知θ就是非連通圖C4m-1∪C12m-8∪G的缺k+1標(biāo)號值的優(yōu)美標(biāo)號.

引理1 對于任意正整數(shù)n,設(shè)C4n是有4n個頂點的圈,則C4n存在特征為2n-1且缺3n的交錯標(biāo)號.

證明記圈C4n上的頂點依次為v1,v2,…,v4n,定義圈C4n的頂點標(biāo)號θ為

容易驗證,θ就是圈C4n的特征為2n-1且缺3n的交錯標(biāo)號.

注意到3n=(2n-1)+n+1,由定理和引理1有如下結(jié)論.

推論1 對于任意正整數(shù)m,非連通圖C4m-1∪C12m-8∪C24m-16存在缺標(biāo)號值12m-8的優(yōu)美標(biāo)號.

例1 由推論1,非連通圖C7∪C16∪C32存在缺標(biāo)號值16的優(yōu)美標(biāo)號為

31,26,30,27,21,28,29,31;

38,17,37,18,36,19,35,20,34,22,33,23,32,24,47,25,38;

0,55,1,54,2,53,3,52,4,51,5,50,6,49,7,48,8,46,9,45,10,44,11,43,12,42,13,41,14,40,15,39,0.

引理2[3]對于任意正整數(shù)m和任意自然數(shù)r1,r2,…,rm,C4m(r1,0,r2,0,…,rm,0,…,0)存在特征為2m-1且缺3m的交錯標(biāo)號.

注意到3m=(2m-1)+m+1,由定理和引理2有如下結(jié)論.

推論2 對于任意正整數(shù)m和任意自然數(shù)r1,r2,…,r6m-4,非連通圖C4m-1∪C12m-8∪C24m-16(r1,0,r2,0,…,r6m-4,0,0,…,0)存在缺標(biāo)號值12m-8的優(yōu)美標(biāo)號.

例2 由推論2,非連通圖C7∪C16∪C32(1,0,2,0,3,0,4,0,5,0,6,0,7,0,8,0,0,…,0)存在缺標(biāo)號值16的優(yōu)美標(biāo)號為

31,26,30,27,21,28,29,31;

38,17,37,18,36,19,35,20,34,22,33,23,32,24,47,25,38;

0(91),90,1(89,88),87,2(86,85,84),83,3(82,81,80,79),78,4(77,76,75,74,73),72,5(71,70,69,68,67,66),65,6(64,63,62,61,60,59,58),57,7(56,55,54,53,52,51,50,49),48,8,46,9,45,10,44,11,43,12,42,13,41,14,40,15,39,0.

引理3[3]對于任意正整數(shù)m和任意自然數(shù)r,C4m(r,r,…,r)存在特征為2m(r+1)-1且缺3m(r+1)的交錯標(biāo)號.

注意到3m(r+1)=[2m(r+1)-1]+m(r+1)+1,由定理和引理3有如下結(jié)論.

推論3 對于任意正整數(shù)m,n和任意自然數(shù)r,當(dāng)6m-4=n(r+1)時,非連通圖C4m-1∪C12m-8∪C4n(r,r,…,r)存在缺2n(r+1)標(biāo)號值的優(yōu)美標(biāo)號.

例3 由推論3給出的非連通圖C7∪C16∪C16(1,1,…,1)缺標(biāo)號值16的優(yōu)美標(biāo)號為

31,26,30,27,21,28,29,31;

38,17,37,18,36,19,35,20,34,22,33,23,32,24,47,25,38;

0(55),54(1),2(53),52(3),4(51),50(5),6(49),48(7),8(46),45(9),10(44),43(11),12(42),41(13),14(40), 39(15),0(55).

由推論3給出的非連通圖C7∪C16∪C8(3,3,…,3)缺標(biāo)號值16的優(yōu)美標(biāo)號為

31,26,30,27,21,28,29,31;

38,17,37,18,36,19,35,20,34,22,33,23,32,24,47,25,38;

0(55,54,53),52(1,2,3),4(51,50,49),48(5,6,7),8(46,45,44),43(9,10,11),12(42,41,40),39(13,14,15),0(55,54,53).

由推論3給出的非連通圖C7∪C16∪C4(7,7,7,7)缺標(biāo)號值16的優(yōu)美標(biāo)號為

31,26,30,27,21,28,29,31;

38,17,37,18,36,19,35,20,34,22,33,23,32,24,47,25,38;

0(55,54,53,52,51,50,49),48(1,2,3,4,5,6,7),8(46,45,44,43,42,41,40),39(9,10,11,12,13,14,15),0(55,54,53,52,51,50,49).

3 結(jié) 語

本文研究了非連通圖C4m-1∪C12m-8∪G的優(yōu)美標(biāo)號,給出了非連通圖C4m-1∪C12m-8∪G是優(yōu)美圖的一個充分條件,可為繼續(xù)研究非連通圖Cm1∪Cm2∪…∪Cmn∪G的優(yōu)美性提供借鑒.

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