行政院國家科學委員會專題研究計畫 成果報告
多環網路與 Cayley 有向圖的漢米爾頓性質與相關問題(2/2)
計畫類別： 個別型計畫 計畫編號： NSC94-2115-M-110-003- 執行期間： 94 年 08 月 01 日至 95 年 07 月 31 日 執行單位： 國立中山大學應用數學系(所) 計畫主持人： 董立大 計畫參與人員： 林承穎 王鴻志 顏珮嵐 洪蓉婷 李昀叡 吳俞鋒 報告類型： 完整報告 處理方式： 本計畫可公開查詢中 華 民 國 95 年 10 月 24 日
行政院國家科學委員會補助專題研究計畫成果報告
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多環網路與 Cayley 有向圖的漢米爾頓性質與相關問題(2/2)
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計畫類別：個別型計畫
計畫編號：NSC 94－2115－M－110－003－
執行期間：94 年 08 月 01 日至 95 年 07 月 31 日
計畫主持人：董立大
執行單位：中山大學應用數學系
中 華 民 國 94 年 10 月 25 日
行政院國家科學委員會專題研究計畫期中報告
多環網路與 Cayley 有向圖的漢米爾頓性質與相關問題(2/2)
The Hamiltonian Property of Multi-loop Networks and Cayley Digraphs and
Related Problems
計畫編號：NSC 94－2115－M－110－003
執行期限：94 年 8 月 1 日至 95 年 7 月 31 日
主持人：董立大(中山大學應用數學系)
計畫參與人員：林承穎 王鴻志 顏珮嵐 洪蓉婷 李昀叡 吳俞鋒
一、中文摘要 第一個部份，我們考慮多環網路上的 漢米爾頓性質。我們利用(i,j,k)-交換法尋 找多環網路的漢米爾頓圈。 第二個部份，在相關的問題探討上我 們考慮圖的漢米爾頓譜與上漢米爾頓數。 這部份裡我們討論圈(Cycle)上的漢米爾頓 譜與證明 Chartrand, Saenpholphat, Thomas, 及 Zhang 在偶圈的上漢米爾頓數猜測。 第三部份，我們研究二部圖與平面圖 的有向著色。這個部份我們給予完全 k 部 圖有向著色數的值，並給予一個直接的證 明:平面圖 G(5,6)的有向著色至少是 7。 第四部份，我們研討完全 k-部圖的獨 立邊數。這裡我們獲得平衡的完全 k-部圖 及部份不平衡的完全 k-部圖所有可能的獨 立邊數。 關鍵詞: 多環網路，漢米爾頓性質，(i,j,k)-交換，漢米爾頓譜，漢米爾頓數，上漢米 爾頓數，圈，有向著色，獨立邊，k 部圖 Abstract Multi-loop networks, Dn(h1, h2, ..., hm) =(V,A) with V={0,1,2,...,n-1} and E={(i,j):j≡ i+hk (mod n) for k=1,2,...,m}, have been widely investigated as architectures for many local area networks in the last few years. There are many properties that have been studied in interconnection networks. The existence of a hamiltonian circuit for an interconnection network is one of the most important properties. If m=2, then Dn(h1,h2)is called a double-loop network. Fiol and Yebra [8] obtained a necessary and sufficient condition for the Hamiltonian property of Dn(h1,h2). In the part, we study the hamiltonian property of multi-loop networks by (i,j,k)-changes.
Second, we consider the hamiltonian spectra and upper hamiltonian numbers of graphs. We determine the Hamiltonian spectra of cycles and prove the conjecture of Chartrand, Saenpholphat, Thomas, and Zhang on the upper hamiltonian number of an even cycle.
Third, we study the oriented colorings of bipartite graphs and planar graph. We determine the oriented chromatic numbers of complete k-partite graphs and give a direct proof for the oriented chromatic number of G(5,6) at least 7.
Fourth, let i(D) the number of independent arcs in an acyclic digraph D and N(G) = {i(D): D is an acyclic orientation of G}. We determine N(G) for any balanced complete k-partite graph G, showing that N(G) is not a set of consecutive integers.
Keywords: Hamiltonian, multi-loop network,
(i,j,k)-exchange, hamiltonian spectrum, hamiltonian number, upper hamiltonian number, cycle, oriented coloring, independent arc, k-partite graph
二、緣由與目的(Introduction)
Multi-loop networks have been widely investigated as architectures for many local area networks in the last few years. They
were first introduced by Wong and Coppersmith [14]. There are many properties that have been studied in lots of literatures [11]. The existence of a hamiltonian circuit in an interconnection network is a significant measurement. The importance of the Hamiltonian property was discussed by Pradhan [13] in proposing dynamically reconfigurable fault-tolerant. Though there are a lot of Hamiltonian multi-loop networks with outdegree at least three, but there are many nonHamiltonian multi-loop networks with outegree three constructed by Locke and Witte [12]. The main purpose of the paper is to determine the Hamiltonian property of multi-loop networks by (i,j,k)-changes.
A Hamiltonian walk of a graph is a walk containing all vertices. Suppose G is a graph with the vertex set V. Given an order v1,v2,…,vn of all vertices of G. Define
h(v1,v2,…,vn)=d(v1,v2)+d(v2,v3)+…+
d(vn-1,vn)+d(vn,v1) where d(x,y) is the
distance between vertices x and y. Then the Hamiltonian spectrum H(G)={h(v1,v2,…,vn): (v1,v2,…,vn) is an order of vertices of G}. The hamiltonian number of G is the minimum number of H(G) and the upper Hamiltonian number h+(G) of G is the maximum number of H(G). In this part, we investigate the Hamiltonian spectra and upper Hamiltonian numbers of cycles. We determine the Hamiltonian spectra of cycles and prove the conjecture of Chartrand, Saenpholphat, Thomas, and Zhang on the upper Hamiltonian number of an even cycle.
Conjecture: (Chartrand et al. [18])
The upper hamiltonian number of the even cycle C2k is 2k2-2k+2.
Oriented colorings were first studied by Courcelle [21] as a tool for encoding graph orientations by means of vertex labels. Let S be a set of k distinct elements. An oriented k-coloring of an oriented graph D is a mapping f from V(D) to S such that (i) if xy∈ A(D), then f(x)≠ f(y) and (ii) if xy,zt∈A(D) and f(x)=f(t), then f(y) ≠ f(z). The oriented chromatic number χo(D) of an oriented graph
D is defined as the minimum k where there
exists an oriented k-coloring of D. For an undirected graph G, let O (G) be the set of all orientations of G. We define the oriented chromatic number χo(G) of G to be the
maximum of χo( D) over D∈O(G). In this
part, we determine the oriented chromatic number of complete bipartite graphs and complete k-partite graphs. A grid G(m,n) is a graph with the vertex set V(G(m,n))={ (i,j):
1≤ i≤ m,1≤ j≤ n} and the edge set E(G(m,n))={(i,j)(x,y)|(i=x+1andj=y) or(i=x and j=y+1)}. Fertin, Raspaud and Roychowdhury [22] proved χo(G(4,5)³7 by
computer programs. Here, we give a proof of
χo(D(5,6)³7 where D(5,6) is the orientation
of G(5,6).
Suppose D is an acyclic orientation of a graph G. An arc of D is said to be independent if its reversal results another acyclic orientation. Denote i(D) the number of independent arcs in D and N(G) = {i(D): D is an acyclic orientation of G}. Also, let imin
(G) be the minimum of N(G) and imax (G) the
maximum. While it is known that imin
(G)=|V(G)|-1 for any connected graph G, the present paper determines imax(G) for
complete k-partite graphs G. We then determine N(G) for any balanced complete k-partite graph G, showing that N(G) is not a set of consecutive integers. This answers a question of West's in [37]. Finally, we give some complete k-partite graphs G whose N(G) is a set of consecutive integers.
三、結果與討論
1. The Hamiltonicity of Multi-loop Networks
In the study of Hamiltonicity of mul-tiloop networks, we get some results by (i,j,k)-exchanges. Suppose n, h1,h2, …, hm are positive integers. A muti-loop network Dn(h1,h2, …, hm) is a digraph with the vertex set {0,1,2,…,n−1} and the arc set {(i,j): j≡i+hk (mod n) for some k=1,2,…,m}.
Define that S(p1,p2,…,pd)= {p1,p2,…,pd} ∪{n0+n1+…+n2k: k,n0,n2k≥1, n2i: a
nonnegative integer, n2i+1: a positive odd
integer, and n0+n1, n2k-1+n2k, n2j+1+n2j+2+n2j+3 ∈{p1,p2,…,pd} for i=1,2,…,k and j=1, 2, …,
k−2}.
Theorem 1.
Suppose n, b, m, h, a, p1,p2,…,pd are positive
integers with m=gcd(n,h+a), b=n/m>1, and gcd(m,a)>1. Then Dn(h,h+a,h+p1a,h+
p2a, ... , h + pda) is not Hamiltonian.
Theorem 2.
Suppose n, b, m, h, a, p1,p2,…,pd are positive
integers with m=gcd(n,h+a), b=n/m>1 and gcd(m,a)=1. If m∈ S(p1,p2,…,pd), then Dn(h,
h+a,h+p1a,h+p2a,...,h+pda) is Hamiltonian.
Theorem 3.
Suppose n, b, h, and a are positive integers with m=gcd(n,h+a), b=n/m>1 and gcd(m,a)=1. If p is even, then m∈S(p)= {p,p+1,p+3,…,2p-1}∪{q: q≥2p} and Dn(h,
h+a,h+pa) is Hamiltonian for m∈S(p).
Theorem 4.
Suppose n, b, h, and a are positive integers with m=gcd(n,h+a), b=n/m>1 and gcd(m,a) =1. If p is odd, then S(p)={p+2q: q is a nonnegative integer} and Dn(h,h+a,h+pa) is
Hamiltonian for m∈S(p).
Theorem 5.
Suppose n, b, m, h, a, p1,p2,…,pd are positive
integers with m=gcd(n,h+a), b=n/m>1 and gcd(m, a)=1. If p,q∈{p1, p2, … ,pd}with p+ q:
odd, and m≥p>q, then Dn(h,h+a,h+p1a,h
+p2a,...,h+pda) is Hamiltonian.
2. Hamiltonian Spectra
In the studies of hamiltonian spectra of graphs, we have the following results .
Theorem 6.
For an even integer n≥4, H(Cn)=
{n,n+2,…,n2/2-n, n2/2-n+2}.
Theorem 7.
For an odd integer n≥3, H(Cn)=
{n,n+2,…,2n-5,2n-3}∪{2n-2,2n-1,…, n2/2-n/2-2}∪{,n2/2-n/2}.
For the study of upper hamiltonian number, we have:
Theorem 8.
For an integer n≥3, if n is even then h+(Cn)=n2-n+2; otherwise, h+(Cn)=n2-n/2.
3. Oriented Colorings
Theorem 9. Let m,n be positive integers with
m³n. Then the oriented chromatic number of Kn,m is m+n for m<2n or n+2n for m³2n.
Theorem 10. Let p1, p2,…,pk be positive
integers with p1³p2³…³pk. Then the oriented chromatic number of Kp1, p2,…,pk is
p1+p2+…+pk for p1<2 p2+…+pk or p2+…+pk +2 p2+…+pk for p1³2 p2+…+pk.
Fertin, Raspaud, Roychowdhury [22] gave the lower bound 7 of oriented chromatic number of planar graphs by computer programs. Here, we provide a direct proof. We find an orientation D(5,6) of the plane graph G(5,6) such that the oriented chromatic number of D(5,6) is at least 7.
4. The Numbers of Independent Arcs of Acyclic Orientations of Complete k-partite Graphs
Let Kk(n) be the complete k-partite graph with
each partite set having n vertices. Then Kk(n) is
Theorem 11. If n³3, then N(K3(n)) = {2n2-xn-
2y: 0≤x≤1,0≤y≤n-1} ∪ {z:3n-1≤z ≤2n2-2n+ 2 }. If n=2 then N(K3(2))={5,6,8}.
Theorem 12. If k³4 and n³2, then N(Kr(n))
= {(r-1)n2-2y:0≤y≤n-1\}∪{z:rn-1≤z≤ (r-1)n2- n}. 四、計畫成果自評 目前所獲得的結果裡，已有部分整理 成文章投槁，其他的部分也將整理成文章 預備投槁。 五、參考文獻(References)
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