姜海寧

(廈門大學(xué)數(shù)學(xué)科學(xué)學(xué)院,福建廈門 361005)

同階雙軌道連通圖的超圈邊連通性

姜海寧

(廈門大學(xué)數(shù)學(xué)科學(xué)學(xué)院,福建廈門 361005)

對(duì)于圖G,如果G?F是不連通的且至少有兩個(gè)分支含有圈,則稱F為圖G的圈邊割.如果圖G有圈邊割,則稱其為圈可分的.最小圈邊割的基數(shù)叫作圈邊連通度.如果去除任何一個(gè)最小圈邊割,總存在一分支為最小圈,則圖G為超圈邊連通的.設(shè)G=(G1;G2;(V1;V2)) 為V軌道圖,最小度?(G)≥4,圍長g(G)≥6且|V1|=|V2|.假設(shè)Gi是ki-正則的,k1≤k2且G1包含一個(gè)長度為g的圈,則G是超圈邊連通的.

圈邊割;圈邊連通度;超圈邊連通性;軌道

本工作中,規(guī)定?有的圖都是無環(huán)和無重邊的無向連通圖.

互連網(wǎng)絡(luò)拓?fù)渫ǔＲ詿o向圖為數(shù)學(xué)模型,而圖的連通度則是網(wǎng)絡(luò)容錯(cuò)的一個(gè)重要指標(biāo).近年來,由于傳統(tǒng)的連通度的局限性,人們對(duì)連通度的概念進(jìn)行了推廣,如限制性邊連通度和圈邊連通度等[1-5].

對(duì)于無向圖G,如果G?F可以使兩個(gè)圈分離,則稱邊集F為圖G的圈邊割.含有圈邊割的圖稱為圈可分圖.文獻(xiàn)[6-7]刻畫了?有不含分離圈的多重圖,因此本工作考慮到研究圈可分圖的性質(zhì).Plummer[8]定義了圖G的圈邊連通度,并且記為λc(G),λc(G)是G中最小圈邊割的基數(shù).

圈邊連通度在圖論的染色、容錯(cuò)及整數(shù)流等[9-15]經(jīng)典領(lǐng)域中起著重要作用.作為網(wǎng)絡(luò)可靠性的一個(gè)重要指標(biāo),圈邊連通度λc(G)比典型的邊連通度具有更高的實(shí)用性.

對(duì)任意的圈可分圖均有λc(G)≤?(G)(*),其中?(G)=min{ω(X)},X在G中導(dǎo)出最小圈[16].如果λc(G)=?(G),則稱圈可分圖為圈最優(yōu)的,并記為λc-最優(yōu).ω(X)是滿足這一條件的所有邊的個(gè)數(shù),這些邊的頂點(diǎn)需要一端在X中,另一端在V(G)X中.

對(duì)于任意點(diǎn)集X,G[X]是指X在G中的導(dǎo)出子圖.X為X的補(bǔ)集.對(duì)于點(diǎn)集X,Y?V, [X,Y]G為邊集,它的每一條邊一端在X中,另一端在Y中.如果[X,X]是最小的圈邊割,則G[X]和G[X]都是連通的.如果[X,X]是最小的圈邊割,則稱點(diǎn)集X為圈邊片段.對(duì)于基數(shù)最小的圈邊片段,則稱為圈邊原子.如果去掉圖G中任何一個(gè)基數(shù)最小的圈邊割都會(huì)導(dǎo)致其中一個(gè)分支是最小圈,則稱連通圖G是超圈邊連通的,記為超-λ[16]c.

設(shè)X是一個(gè)圈邊片段,如果X和X都導(dǎo)不出最小圈,則稱X為超圈邊片段.基數(shù)最小的超圈邊片段為超圈邊原子[17].在本工作中,用超片段和超原子分別代替超圈邊片段和超圈邊原子.如果圈邊片段可以導(dǎo)出一個(gè)圈,則稱它是平凡的,否則稱為非平凡的.如果X是超片段, 則X也是超片段,且G[X]和G[X]都是連通的.

如果Aut(G)可以在V(G)(E(G))上作用傳遞,則稱圖G是點(diǎn)(邊)傳遞的.設(shè)x∈V(G),則稱集合{xg|g∈Aut(G)}為Aut(G)的一個(gè)軌道.顯然,Aut(G)分別在每個(gè)軌道上作用傳遞.設(shè)圖G=(G1,G2,(V1,V2))為連通圖,如果Aut(G)分別在V1和V2上作用傳遞且|V1|=|V2|,則稱圖G為同階雙軌道連通圖.傳遞圖具有高容錯(cuò)性、傳遞延遲等[18-21]屬性,因此在刻畫網(wǎng)絡(luò)拓?fù)浣Y(jié)構(gòu)時(shí)起著重要作用.

Zhou[22]刻畫了圈邊連通度條件下圖的原子.Wang等[16]指出對(duì)于任意的最小度δ(G)≥4 且|V|≥6的邊傳遞圖是圈最優(yōu)的.在圖的圍長g≥5的條件下,k(≥4)-正則點(diǎn)傳遞圖是λc-最優(yōu)的.Wang等[16]和Zhang等[17]指出最小度δ(G)≥4且|V|≥6的邊傳遞圖是λc-最優(yōu)的.本工作將證明圍長≥6的條件下階數(shù)相同的雙軌道連通圖是超圈邊連通的.

1 引理

?理1[17]一個(gè)圈最優(yōu)圖不是超圈邊連通的充要條件是它有超原子.

2 同階雙軌道圖的超-λc5

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Super cyclically edge connected graphs with two orbits of the same size

JIANG Haining
(School of Mathematical Sciences,Xiamen University,Xiamen 361005,Fujian,China)

For a graph G,an edge set F is a cyclic edge-cut if(G?F)is disconnected and at least two of its components contain cycles.If G has a cyclic edge-cut,it is said to be cyclically separable.The cyclic edge-connectivity is cardinality of a minimum cyclic edgecut of G.A graph is super cyclically edge-connected if removal of any minimum cyclic edge-cut makes a component a shortest cycle.Let G=(G1;G2;(V1;V2))be a doubleorbit graph with minimum degree?(G)≥4,girth g≥6 and|V1|=|V2|.Suppose Giis ki-regular,k1≤k2and G1contains a cycle of length g,then G is super cyclically edge connected.

cyclic edge-cut;cyclic edge-connectivity;super cyclically edge-connected; orbit

1007-2861(2017)02-0252-05

10.3969/j.issn.1007-2861.2016.05.004

2016-03-23

國家自然科學(xué)基金資助項(xiàng)目(11471273)

通信?者:姜海寧(1983|),男,博士研究生,研究方向?yàn)閳D論.E-mail:hnjiangsd@163.com