The path partition number of a graph by ELIZABETH JONCK THESIS submitted in the fulfilment of the requirements for the degree PHILOSOPHIAE DOCTOR in MATHEMATICS in the FACULTY OF SCIENCE at the RAND AFRIKAANS UNIVERSITY PROMOTER: PROF. I. BROERE October 1997 Acknowledgements I wish to thank: • RAU for their financial assistance, • Professor Izak Broere for his motivation and instructive guidance, • Cobus and Lindy for their professionalism and friendly help, • Andries for his encouragement and help with the proofreading. I wish to dedicate this thesis to my family, in particular to "Pa Frik" and "Ma Mart"; Your love and support are precious to me! Contents 1 Introduction and overview 1 1.1 Introduction ....................................1 1.2 Overview ......................................2 2 The induced path number of some classes of graphs 4 2.1 The induced path number of bipartite graphs .................4 2.1.1 Introduction ................................4 2.1.2 The induced path number of some classes of bipartite graphs . 6 2.2 The induced path number of a complete multipartite graph ..........14 2.2.1 Introduction ................................14 2.2.2 Proof of Theorem 2.17 ..........................16 I 3 The induced path number of some products of some graphs 31 3.1 Introduction ....................................31 3.2 Products of complete graphs ...........................32 3.2.1 The induced path number of Km X K ................ 32 3.2.2 The induced path number of the complement of Km X K ..... 42 3.2.3 The induced path number of Cm X Cn .................49 4 Uniquely partitionable and saturated graphs with respect to linear ar- boricity 56 4.1 Introduction ....................................56 4.2 Uniquely partitionable and saturated graphs ..................57 4.3 The order of uniquely partitionable graphs ...................62 5 Vertex and edge addition and deletion 65 5.1 The path partition number and the addition or deletion of a vertex or an edge. 65 5.2 p-criticality ....................................68 Bibliography 75 Index 78 List of Figures 2.1 An induced path partition 6 2.2 The binary trees B2 and 133 ...........................7 2.3 The cartesian product C1 x C2 .........................9 2.4 The 2-dimensional mesh Md,,d2 .........................9 2.5 The butterflies B1 , B2 and B3 12 2.6 Partition classes of cardinality one, two and three ...............17 2.7 Partition classes containing a vertex of V€ .................... 18 2.8 Partition classes containing one or two vertices of V1 .............. 20 2.9 The remaining subcases of case (iii) .......................21 2.10 The remaining subcases of case (viii) ......................22 2.11 The situation after y iterations .........................23 2.12 The remaining subcase of case (ix) .......................24 111 iv 2.13 The remaining subsubcase of case (ix) ..................... 24 2.14 The situation after z iterations ......................... 25 2.15 The seventh case ................................. 27 3.1 Rows and columns of C ............................. 32 3.2 Km x K, m odd (the first case) ......................... 34 3.3 If, x K, in even (the first case) ........................ 35 3.4 Km x K,, in a multiple of 4 (the second case) ................. 37 3.5 Kn x K,, in not a multiple of 4 (the second case) ............... 38 3.6 Km x K, m odd (the third case) ........................ 40 3.7 K, x If,,, m even (the third case) ........................ 41 3.8 E3 x ....................................... 42 3.9 K3 XIc, n odd .................................. 45 3.10 K3 X K, n even ................................. 46 3.11 <V1> ....................................... 3.12 Cm X C,-, fl = m .................................. 51 3.13 Cm X C, fl > m ................................. 52 3.14 Cm X C, n 4a and m = 2a(2k - 1) + 1 .................... 53 V 3.l5CmXCn ,m even and n>in 55 4.1 Possible partitions ................................61 4.2 Possible partitions ................................63 4.3 (a) Graph F (b) <Vm U Vi > ..........................64 5.1 C3.K5 ...................................... 70 Opsomming Die gemnduseerde padgetal p(G) van 'n grafiek C word gedefinieer as die minimum aantal deelversamelings waarin die puntversameling V(G) van G verdeel kan word so dat elke deelversameling 'n pad induseer. Ons bepaal die geinduseerde padgetal van 'n volledige £-ledige grafiek. Verder ondersoek ons die geInduseerde padgetal van produkte van volledige grafieke, van die komplement van sulke produkte en van produkte van sikiusse. Vir 'n grafiek C word die lineêre puntbosagtigheid Iva(G) gedefinieer as die minimum aantal deelversamelings waarin die puntversameling van C verdeel kan word so dat elke deelver- sameling 'n lineêre bos induseer. Omdat elke pad 'n lineêre bos is, is Iva(G) p(G) vir elke grafiek G. Ons noem 'n grafiek C uniek rn-lineér-bos-verdeelbaar as Iva(G) = in en daar slegs een verdeling van V(C) in in deelversamelings is so dat elke deelversameling 'n lineêre bos induseer. As Iva(G) in en lva(G+e) > in vir elke e E E(), sê ons die grafiek C is rn-lva- versadig. Ons konstrueer grafieke wat uniek m-lineêr-bos-verdeelbaar en m-lva-versadig is. Ons karakteriseer die ordes van uniek m- pad- verdeel bare onsamehangende, samehangende en in-p-versadigde grafleke. vi vii Ons bekyk die invloed van byvoeging en weglating van 'n punt of 'n lyn op die geInduseerde padgetal. As C 'n grafiek is so dat p(G) = k en p(G - v) = k - 1 vir elke v E V(G), dan sê ons dat C k-minus-kritiek is. Ons bewys dat as C 'n samehangende grafiek is, bestaande uitsikliese blokke Bi met p(B1 ) = b1 yin = 1,2, ... ,n waar n >2 en k= b1 —n+ 1, dan is C k-minus-kritiek as en slegs as elke blok B 'n b1-minus-kritieke grafiek is. Summary The induced path number p(G) of a graph G is defined as the minimum number of subsets into which the vertex set V(G) of G can be partitioned such that each subset induces a path. In this thesis we determine the induced path number of a complete £-partite graph. We investigate the induced path number of products of complete graphs, of the complement of such products and of products of cycles. For a graph G, the linear vertex arboricity lva(G) is defined as the minimum number of subsets into which the vertex set of C can be partitioned so that each subset induces a linear forest. Since each path is a linear forest, Iva(G) p(G) for each graph C. A graph G is said to be uniquely rn-li near- forest- partition able if lva(C) = in and there is only one partition of V(G) into m subsets so that each subset induces a linear forest. Furthermore, a graph C is defined to be nz- Iva- saturated if Iva(G) < in and lva(C + e) > iii for each e E We construct graphs that are uniquely n2-linear-forest-partitionable and in-lva-saturated. We characterize those graphs that are uniquely m-linear-forest-partitionable and rn-lva- saturated. We also characterize the orders of uniquely in- path- partitionable disconnected, connected and rn-p-saturated graphs. viii ix We look at the influence of the addition or deletion of a vertex or an edge on the path partition number. If C is a graph such that p(G) = k and p(G - v) = k - 1 for every v E V(G), then we say that C is k-minus-critical. We prove that if C is a connected graph consisting of cyclic blocks Bi with p(B1 ) = b, for i = 1,2, ... ,n where ii > 2 and k bi - n+ 1, then C is k- minus- critical if and only if each of the blocks B1 is a bj- minus- critical graph. Gliapter 1 Introduction and overview 1.1 Introduction We generally use the notation and terminology of [8]. For a graph C, the induced path number p(G) is defined by Chartrand e.a. in [9] as the minimum number of subsets into which the vertex set V(C) of G can be partitioned such that each subset induces a path. They investigated the induced path number for bipartite graphs and presented formulas for the induced path number of complete bipartite graphs and complete binary trees. They also determined the induced path number of all trees and considered the induced path numbers of meshes, hypercubes and butterflies. 1 1.2 Overview In Chapter 2 we summarise the results of [9] and determine the induced path number of a complete £-partite graph. We investigate the induced path number of products of complete graphs, of the complement of such products and of products of cycles in Chapter 3. For a graph C, the linear vertex arboricity Iva(G) is defined as the minimum number of subsets into which the vertex set of C can be partitioned so that each subset induces a linear forest. A graph G is said to be uniquely rn-linear-forest-partitionable if Iva(G) = in and there is only one partition of V(G) into m subsets so that each subset induces a linear forest. Furthermore, a graph C is defined to be rn-lva-saturated if Iva(G) in and lva(G + e) > in for each e E(). In Chapter 4 we construct graphs that are uniquely rn-linear-forest- partitionable and m-lva-saturated. We characterize those graphs that are uniquely rn-linear- forest-partitionable and rn- Iva- saturated. We also characterize the orders of uniquely in- path-partitionable disconnected, connected and rn-p-saturated graphs. In Chapter 5 we look at the influence of the addition or deletion of a vertex or an edge on the path partition number. If C is a graph such that p(C) = k and p(G - v) = k - 1 for every v E V(C), then we say that C is k-minus-critical.