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博碩士論文 etd-0725106-232541 詳細資訊
Title page for etd-0725106-232541
論文名稱
Title
二部圖的定向著色
The Oriented Colourings of Bipartite Graphs
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
24
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2006-07-21
繳交日期
Date of Submission
2006-07-25
關鍵字
Keywords
定向著色數
oriented chromatic number
統計
Statistics
本論文已被瀏覽 5763 次,被下載 2119
The thesis/dissertation has been browsed 5763 times, has been downloaded 2119 times.
中文摘要
令S是一個包含k個不同元素的集合。若一個函數f從V(D)對應到S使得(i)如果xy是D上的一條定向邊,則f(x)≠f(y),及(ii)如果xy和zt是D上的兩條定向邊以及f(x)=f(t),則f(y)≠f(z);則此f稱為一個定向圖D上的一個定向k著色。在一個定向圖D上,若存在一個定向k著色且k是最小,則此k即為D的定向著色數,記為Xo(D)。令O(G)是包含無向圖G上所有定向圖的集合。若D是O(G)的一個元素且使得Xo(D)為最大,則此Xo(D)即為G的定向著色數,記為Xo(G)。在這篇文章中,我們討論了完全二部圖和完全k部圖的定向著色數。令一個格子圖G(m,n)的點集合為V(G(m,n))={(i,j)|1≦i≦m,1≦j≦n} 且邊集合為 E(G(m,n))={(i,j)(x,y) | (i=x+1 and j=y) or (i=x
and j=y+1)}。 Fertin,Raspaud 和 Roychowdhury [3] 利用電腦程式證明Xo(G(4,5))≧7。在此,我們給予一個G(5,6)上的某個定向圖D(5,6),使得Xo(D(5,6)}=7的一個證明。
Abstract
Let S be a set of k distinct elements. An oriented k coloring of an oriented graph D is a mapping f:V(D)→S such that (i) if xy is conatined in A(D), then f(x)≠f(y) and (ii) if xy,zt are conatined in
A(D) and f(x)=f(t), then f(y)≠f(z). The oriented chromatic
number Xo(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 D of G. We
define the oriented chromatic number Xo(G) of G to be the
maximum of Xo(D) over D conatined by O(G). In this thesis, 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+1 and j=y) or (i=x and j=y+1)}. Fertin, Raspaud and Roychowdhury [3] proved Xo(G(4,5))≧7 by computer programs. Here, we give a proof of
Xo(D(5,6)=7 where D(5,6) is the orientation of
G(5,6).
目次 Table of Contents
1 Introduction 2
2 Previous results 4
3 The main results 7
4 Conclusion 16
參考文獻 References
[1] O.V. Borodin, A.V. Kostochka, J. Neˇsetˇril, A. Raspaud, E. Sopena, On the maximum
average degree and the oriented chromatic number of a graph, Discrete Math.
206 (1999) 77-89.
[2] B.Courcelle. The monadic second order logic of graphs VI: On several representations
of graphs by relational structures, Discrete Applied Math. 54 (1994), 117-149.
[3] G. Fertin, A. Raspaud, A. Roychowdhury, On the oriented chromatic number of
grids, Inf. Process. Lett. 85 (2003) 261-266.
[4] A.V. Kostochka, E. Sopena, X. Zhu, Acyclic and oriented chromatic number of
graphs, J. Graph Theory 24 (4) (1997) 331-340.
[5] P. Ochem, Oriented colorings of triangle-free planar graphs, Inf. Process. Lett. 92
(2004) 71-76.
[6] A. Raspaud and E. Sopena, Good and semi-strong colorings of oriented planar
graphs, Inf. Process. Lett. 51 (1994), 171-174.
[7] E. Sopena, The chromatic number of oriented graphs, J. Graph Theory 25 (1997)
191-205.
[8] E. Sopena, There exist oriented planar graphs with oriented chromatic number at
least sixteen, Inf. Process. Lett. 81 (6) (2002) 309-312.
[9] A. Szepietowski, M, Targan. A note on the oriented chromatic number of grids, Inf.
Process. Lett. 92 (2004) 65-70.
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