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Odd Sums of Long Cycles in 2-Connected Graphs

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Odd Sums of Long Cycles in 2-Connected Graphs Odd Sums of Long Cycles in 2-Connected Graphs by Cong Teng A Thesis Submitted to the Faculty of The Charles E. Schmidt College of Science in Partial Fulfillment of the Requirements of the Degree of Master of Science Florida Atlantic University Boca Raton, Florida August 1999 Odd Sums of Long Cycles in 2-Connected Graphs by Cong Teng This thesis was prepared under the direction of the candidate's thesis advisor, Dr. Stephen C. Locke, Department of Mathematics, and has been approved by the members of her supervisory committee. It was submitted to the faculty of The Charles E. Schmidt College of Science and was accepted in partial fulfillment of the requirements for the degree of Master of Science. SUPERVISORY COMMITTEE: ~c~ tL~~ Chairman, Department of Mathematics Dean of Graduate Studies and Research 11 Acknowledgement First, I want to give my deepest thanks to my advisor, Dr. Stephen Locke, for his guidance, inspiration and encouragement through my study and mas­ ter's thesis process. I would also like to give my high appreciation to the committee member, Dr. Fred Richman, for his advice, strict training in Ab­ stract Algebra, and especially his attitude toward research and life which has some effects on me. I feel deeply thankful to Dr. Heinrich Niederhausen, who is also in my committee, for his support and kind encouragement. Also Mr. Volker Leek, I deeply appreciate your help in computer skill for ad­ justing and formatting this thesis. I would like to thank all the faculty in mathematics department who have directly or undirectly supported my life at FAU. I would like to give special thanks to Dr. Yuan Wang for her warm support both as a teacher and a friend. lll Abstract Author: Cong Teng Title: Odd Sums of Long Cycles in 2-connected graphs Institution: Florida Atlantic University Thesis Advisor: Dr. Stephen C. Locke Degree: Master of Science Year: 1999 Let G be a 2-connected graph with minimum degree d and with at least d + 2 vertices. Suppose that G is not a cycle. Then there is an odd set of cycles, each with length at least d + 1, such that they sum to zero. If G is also non-hamiltonian or bipartite, cycles of length at least 2d can be used. lV Table of Contents 1. INTRODUCTION 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2. NOTATION AND TERMINOLOGY ... .................................. 5 3. 2d-KERNELS OF 2-CONNECTED GRAPHS ... ... ..... .......... ..... 7 4. (d + 1)-KERNELS OF 2-CONNECTED GRAPHS .......... .. .... ..... 29 REFERENCES 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 31 v Section 1. Introduction The following two theorems appear in Dirac [7], [8]. Theorem 1.1 (Dirac, [7]) Let G be a 2-connected graph with minimum degree d and at least 2d vertices. Then G contains a cycle of length at least 2d. Theorem 1.2 (Dirac, [8]) Let G beak-connected graph and X be a set of k vertices of G, k 2: 2. Then G contains a cycle C which contains every vertex of X. The cycle space of a graph G is the vector space spanned by edge sets of cycles of G over G F2 = { 0, 1}. A set of cycles C is a k-kemel if (i) every cycle inC has length at least k, (ii) ICI is odd, and (iii) every edge of G occurs in an even number of cycles of C. In other words, C is an odd set of cycles, each with length at least k , such that the cycles inC sum to zero. Other standard graph-theoretic terminology will be reviewed in Section 2. Bondy and Lovasz [5] proved that cycles through a specified set of vertices 1 generate the cycle space of a graph. Theorem 1.3 ([5]) Let S be a set of k- 1 vertices in a k-connected graph G. Then the cycles through S generate the cycle space of G Based on Theorem 1.1 and Theorem 1.3, Bondy proposed the following conjecture. Conjecture (Bondy, [3] ) Let G be a 3-connected graph with minimum degree at least d and at least 2d vertices. Then every cycle is the sum of an odd number of cycles of length at least 2d - 1. Locke [11], [13] partially proved this conjecture in the following Theorems. Theorem 1.4 (Locke, [11]) If G is a non-hamiltonian, 3-connected graph with minimum degree d, then every cycle is the sum of cycles of length at least 2d- 1. Theorem 1.5 (Locke, [13]) Let G be a 2-connected non-hamiltonian graph with minimum degree d. Then G has a 2d-kernel. 2 Theorem 1.4 and Theorem 1.5 combine to yield Theorem 1.6. Theorem 1.6 If G is a non-hamiltonian, 3-connected graph with minimum degree d, then every cycle is the sum of an odd number of cycles of length at least 2d- 1. Clearly, if G is (2d- 1)-generated and there are an odd number of cycles summing to zero, each with length at least 2d- 1, then every cycle is the sum of an odd number of cycles, each with length at least 2d- 1. In [9], [10], Hartman gave the following result. Theorem 1. 7 (Hartman, [9], [10]) Let G be a 2-connected graph with min­ imum degree d, where G is not Kd+l if d is odd. Then the cycles of length at least d + 1 generate the cycle space of G. As an extension to Theorem 1.7, we prove the following two theorems. These results have been submitted to Discrete Mathematics [14]. Theorem 1.8. Let G be a 2-connected graph with minimum degree d, where G is not Kd+l if dis not odd. Then G has a (d + 1)-kernel. 3 Combining Theorem 1.7 and Theorem 1.8, we obtain the following theo­ rem. Theorem 1.9 Let G be a 2-connected graph with rninimillil degree d and at least d + 2 vertices. Then every cycle is the sum of an odd nlUilber of cycles of length at least d + 1. We also give a result which is a little closer to Bondy's Conjecture. Theorem 1.10 Let G be a 2-connected hamitonian graph with minimum degree d at least 3. Suppose G contains no cycle of length v( G) - 1 or no cycle of length v( G) - 2. Then G has a 2d-kernel. 4 Section 2. Notation and Terminology All graphs considered in this thesis are simple graphs. Notation and terminology not defined in this thesis can be found in [6]. Let G be a graph. We denote by v( G) and c-( G) the number of vertices and the number of edges of G , respectively. The degree of vertex v in G is dc(v) or d(v). 8(G) and .6.(G) are the minimum and maximum degrees of vertices in G. The length of a path or a cycle is the cardinality of its edge set. We use de( u, v) or d(u, v) for the distance from u to v, and P(u, v) the segment of P from u to v. A graph G is connected if every pair of vertices is connected by a path. G is called k-connected (for k E N) if IGI 2:: k, and G- X is connected for every set X~ V with lXI < k. If IGI > 1 and G- F is connected for every set F ~ E of fewer than l edges, then G is called l-edge-connected. The cycle space of a graph G is the vector space of edge sets of G over GF(2). A graph is k-generated if its cycle space is generated by cycles of length at least k. A graph is k-path-connected if, for every pair of vertices x and y, there is an (x, y)-path of length at least k. We call a 2-connected graph a k-generator if G is k-generated and (k- I)-path-connected. For a vertex v, N(v) is the set of neighbors of v in G. For a vertex v and subset X of G, Nx(v) is the set of neighbors of v in X, that is, Nx(v) = N(v) n V(X). We set nx(v) = INx(v)l. A cut vertex of G is a vertex x such that G-x has more 5 components than G. A connected graph is separable if it has a cutvertex. A maximal connected subgraph without a cutvertex is called a block. A block of G which contains exactly one cut vertex is called an endblock. An internal vertex of an endblock is a vertex of the block different from its cutvertex. vVhen it is necessary, we will assign a direction to a cycle C. In this case, we use C for the opposite direction. Similarly, P denotes the path P travelled in the reverse direction.
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