給定一具有n個點及m個邊的圖G。假設圖G 的最大的二部誘導子圖具有n′ 個點，最大的二部生成子圖具有m′ 個邊，定義G 的二部密度為b(G) = m′/m 及二部比率為b∗(G) = n′/n。參考文獻Zhu [18] 證明了下面結果，除了7 個特殊圖之外，所有不含K3 為子圖的2-連通次3 正則圖G，滿足b(G) ≥ 17/21。除了Petersen 圖及正十二面體之外，所有不含K3 為子圖的2-連通次3 正則圖G 滿足b∗(G) ≥ 5/7。這兩個結果可以由一個技術性的結果得到︰若G 是一個不含K3 為子圖且最小度為2 的2-連通次3 正則圖G，H 為G 中一個最大的二部誘導子圖，則|V (H)| ≥ (5n3 +6n2 + ǫ(G))/7，其中ni = ni(G) 表示度為i 的頂點個數，ǫ(G) ∈ {−2,−1, 0, 1}。為了計算ǫ(G) 的值，需要構造四個圖類： G1, G2, G3 及F-cycles。對於G ∈ Gi，我們有ǫ(G) = −3 + i，若G ∈ F-cycle，則ǫ(G) = 0，其他為ǫ(G) = 1。構造這四個圖類，需要用到11 構圖運算方法。在本論文中，我們將探討圖類G1, G2, G3 的幾何結構特徵。首先借助計算機構造出G1, G2, G3。令P 為包含所有不含K3 為子圖且最小度為2 的2-邊連通次3 正則平面圖的圖類。圖類G′1 收集所有G ∈ P 且每一面的長度為5 的圖，圖類G′2 收集所有G ∈ P 除了一個面的長度為7，其他面的長度為5 的圖，圖類G′3 收集所有G ∈ P 除了一個面的長度為9 或是兩個面的長度為7，其他面的長度為5 的圖。計算機演算的結果顯示對i = 1, 2, 3，Gi ⊆ G′i。在本論文中，我們證明當i = 1, 2 時，Gi = G′i；當i = 3時，我們有G′3 = G3 ∪R，其中R 是一包含9 個F-cycles 的圖類。

Suppose G is a graph with n vertices and m edges. Let n′ be the maximum number of vertices in an induced bipartite subgraph of G and let m′ be the maximum number of edges in a spanning bipartite subgraph of G. Then b(G) = m′/m is called the bipartite density of G, and b∗(G) = n′/n is called the bipartite ratio of G. It is proved in [18] that if G is a 2-connected triangle-free subcubic graph, then apart from seven exceptional graphs, we have b(G) ≥ 17/21. If G is a 2-connected triangle-free subcubic graph, then b∗(G) ≥ 5/7 provided that G is not the Petersen graph and not the dodecahedron. These two results are consequences of a more technical result which is proved by induction: If G is a 2-connected triangle-free subcubic graph with minimum degree 2, then G has an induced bipartite subgraph H with |V (H)| ≥ (5n3 + 6n2 + ǫ(G))/7, where ni = ni(G) are the number of degree i vertices of G, and ǫ(G) ∈ {−2,−1, 0, 1}. To determine ǫ(G), four classes of graphs G1, G2, G3 and F-cycles are onstructed.
For G ∈ Gi, we have ǫ(G) = −3 + i and for an F-cycle G, we have ǫ(G) = 0. Otherwise, ǫ(G) = 1. To construct these graph classes, eleven graph operations are used. This thesis studies the structural property of graphs in G1, G2, G3. First of all, a computer algorithm is used to generate all the graphs in Gi for i = 1, 2, 3. Let P be the set of 2-edge connected subcubic triangle-free planar graphs with minimum degree 2. Let G′
1 be the set of graphs in P with all faces of degree 5,
G′2 the set of graphs in P with all faces of degree 5 except that one face has degree 7, and G′3 the set of graphs in P with all faces of degree 5 except that either two faces are of degree 7 or one face is of degree 9. By checking the graphs generated by the computer algorithm, it is easy to see that Gi ⊆ G′i for i = 1, 2, 3. The main results of this thesis are that for i = 1, 2, Gi = G′i and G′3 = G3 ∪R, where R is a set of nine F-cycles.

1 Introduction 10
2 Graph operations 13
3 Bipartite density of graphs in Q 18
4 Proof of Theorem 2.6 21
5 Proof of Theorem 2.7 and Theorem 2.8 23
6 Algorithm 33
References 37
Appendix 39

[1] M. Albertson, B. Bollobas and S. Ticker, The independence ratio and the maximum degree of a graph, Proceedings of the Seventh Southeastern Conference on Combinatorics, Graph Theory and Computing (Louisiana State University, Baton Rouge, La., 1976), pp. 43-50. Congressus Numerantium, No. XVII, Utilitas Math., Winnipeg, Man., 1976.
[2] B. Bollobás and A. D. Scott, Problems and results on judicious partitions, Random Structure and Algorithm, 21 (2002) 414-430.
[3] J. A. Bondy and S. C. Locke, Largest bipartite subgraphs in triangle-free graphs with maximum degree three, J. Graph Theory 10 (1986) 477-504.
[4] P. ErdH{o}s, Problems and results in graph theory and combinatorial analysis, Graph Theory and Related Topics (Proceedings of Waterloo Conference, dedicated to W. T. Tutte on his sixties birthday, Waterloo, 1977), Academic, New York (1979) 153-163.
[5] S. Fajtlowicz, On the size of independent sets in graphs, Cong. Numerantium 21 (1978), 269-274.
[6] J. Griggs and O. Murphy, Edge density and independence ratio in triangle-free graphs with maximum degree three, Discr. Math. 152 (1996), 157-170.
[7] H. Hatami and X. Zhu, The fractional chromatic number of graphs of maximum degree at most three, Preprint, (2006).
[8] C. Heckman and R. Thomas, A new proof of the independence ratio of triangle-free cubic graphs, Discrete Math. 233 (2001), no. 1-3, 233-237.
[9] G. Hopkins and W. Staton, Extremal bipartite subgraphs of cubic triangle-free graphs, J. Graph Theory 6 (1982) 115-121.
[10] K. F. Jones, Size and independence in triangle-free graphs with maximum degree three, J. Graph Theory 14 (1990), 525-535.
[11] S. C. Locke, Maximum k-colourable subgraphs, J. Graph Theory 6 (1982) 123-132.
[12] G. I. Orlova and Y. G. Dorfman, Finding the maximal cut in a graph, Eng. Cyber. 10 (1972) 502-506.
[13] W. Staton, Edge deletions and the chromatic number, Ars Combinatoria 10 (1980) 103-106.
[14] W. Staton, Some Ramsey-type numbers and the independence ratio, TRans. Amer. Math. Soc. 256 (1979), 353-370.
[15] B. Xu and X. Yu, Triangle-free subcubic graphs with minimum bipartite density, Journal of Combinatorial Theory Ser. B, to appear.
[16] M. Yannakakis, Node- and Edge-Deletion NP-Complete Problems, STOC (1978) 253-264.
[17] X. Zhu, Bipartite density of triangle-free subcubic graphs, Preprint (2007).
[18] X. Zhu, Bipartite subgraphs of triangle-free subcubic graphs, Journal of Combinatorial Theory Ser. B, to appear.