Fakltät für Mathematik, Informatik nd Natrissenschaften der Rheinisch-Westfälischen Technischen Hochschle Aachen arxiv:1711.03464v1 [math.co] 9 Nov 2017 Master Thesis The Strong Colors of Floers The Strctre of Graphs ith chordal Sqares Sebastian Wiederrecht at Lehrsthl II für Mathematik 11.03.2016 Referee: Prof. Dr. Eberhard Triesch Second Referee: Dr. Robert Scheideiler
Abstract A proper vertex coloring of a graph is a mapping of its vertices on a set of colors, sch that to adjacent vertices are not mapped to the same color. This constraint may be interpreted in terms of the distance beteen to vertices and so a more general coloring concept can be defined: The strong coloring of a graph. So a k- strong coloring is a coloring here to vertices may not have the same color if their distance to each other is at most k. The 2-strong coloring of the line graph is knon as the strong edge coloring. Coloring the kth poer G k of a graph G is the same as finding a k-strong coloring of G itself. In order to obtain a graph class on hich the 2-strong coloring becomes efficiently solvable e are looking for a strctre that prodces indced cycles in the sqare of G, so that by exclding this strctre e obtain a graph class ith chordal sqares, here a chordal graph is a graph ithot any indced cycles of length 4. Sch a strctre is called a floer. Another strctre ill be fond and explained, hich is responsible for floers to appear in the line graph of G: The sprots. With this graphs ith chordal line graph sqares are described as ell. Some attempts in generalizing those strctres to obtain perfect graph sqares are being made and the general concept of chordal graph poers, i.e. the existence of a smallest poer for hich a graph becomes chordal, the poer of chordality is introdced in order to solve some coloring related N P-hard problems on graphs ith parameterized algorithms. Some connections to the famos parameter treeidth arise alongside ith some deeper connections beteen edge and vertex coloring. i
Acknoledgements This thesis as the greatest challenge I had to face in my academical life so far and its completion old not have been possible ithot the help and spport of several people on hom I cold and have heavily relied. Before everyone else Robert Scheideiler has to be mentioned not only as the second referee of this thesis bt as my mentor in these early steps into the orld of research. As I move on I definitely ill miss or sometimes hor-long conversations not only abot mathematics bt abot life and chance. Thank yo Robert, for yor spport, yor ideas and even yor criticism. Withot yo this thesis old not exist. In varios forms I received help and spport from friends and especially one stands ot before everyone else, Robert Lö. There are times in hich I actally qestion if I ere able to achieve my bachelor s degree in the first place if it as not for Robert. He is a dear and ise friend ho alays seemed to have time to listen to the problems I had ith my research and he patiently listened going on abot varios topics concerning this thesis even if I cold not come p ith a specific qestion to ask. Together ith Leon Eikelmann, Nora Lüpkes and Pascal Vallet, he endred the tiring act of proof reading. I ish to thank all for of them for dedicating their time on this task. At last my parents deserve some thankfl ords for both their moral and financial spport dring my hole time at the RWTH. They tried to listen and nderstand hat I am doing and alays believed I myself kne hat that as, even if I myself as not entirely sre of it. iii
Complexity is the prodigy of the orld. Simplicity is the sensation of the niverse. Behind complexity, there is alays simplicity to be revealed. Inside simplicity, there is alays complexity to be discovered. Y Gang, Chinese University of Hong Kong v
Contents 1 Introdction 3 2 Definitions and Graph Coloring 5 2.1 Basic Terminology and Definitions....................... 5 2.2 Vertex Coloring................................. 9 2.3 Edge Coloring.................................. 12 3 Poers of Graphs 19 3.1 Upper Bonds on the Chromatic Nmber of Graph Poers......... 20 3.2 Graph Classes Closed Under Taking Poers................. 22 3.2.1 Chordal Graphs............................. 26 3.3 Chordal Poers and Poers of Chordal Graphs................ 30 3.3.1 Chordal Sqares............................. 30 4 Strong Edge Coloring 43 4.1 Introdction................................... 43 4.2 Sqares of the Line Graph........................... 46 4.2.1 The Line Graph............................. 46 4.2.2 Chordality of the Sqared Line Graph................. 48 4.2.3 A Look Into Perfection......................... 77 5 Compting Strong Colorings 83 5.1 Parameterized Complexity........................... 83 5.1.1 Introdction to Complexity Theory.................. 84 5.1.2 Treeidth and Vertex Cover...................... 93 5.2 Chordality.................................... 101 5.2.1 An Algorithmic Approach on Chordal Graphs............. 102 5.2.2 Chordal Poers and Complexity.................... 115 5.2.3 Cliqes in Poers of Graphs...................... 117 6 Conclsion 125 Bibliography 129 1
Chapter 1 Introdction Dividing a given set into sbsets is a fndamental procedre in mathematics and often the sbsets are reqired to satisfy some prescribed reqirements. The argably most poplar division of a set into sbsets in graph theory is the coloring of the vertices of a graph sch that no to adjacent vertices have the same color. With its roots in the nmeros attempts to solve the famos For Color Problem the theory of vertex coloring has become one of the most stdied field of graph theory. And de to the sheer nmber of nsolved problems, open qestions and knon - and nknon - applications, it still is an exciting part of modern science. A nmber of graph coloring problems have their roots in a practical problem in commnications knon as the Channel Assignment Problem. In this problem there are transmitters like antennas or satellite dishes located in some geographic region. Those transmitters may interfere ith each other hich is a very common problem. There is a sheer nmber ob different possible reasons for this interference sch as their distance to each other, the time of day, month or year, the terrain on hich the transmitter is bilt, its poer or the existence of poer lines in the area. Sch a problem can be modeled by a graph hose vertices represent the transmitters and to transmitters are joined by an edge if and only if they interfere ith each other. The goal is to assign freqencies or channels to the transmitters in a manner that prevents the signal of being disrpted by the interference. Obviosly, as this is a real orld problem, there are some restricted resorces like a limited nmber of available channels or a limited bdget to by sch freqencies. This gives some natral optimality criteria in hich the channels shold be assigned. This problem, in nmeros variations, has been stdied in economical and military backgronds, sometimes jst for the sake of its beaty, and, since the interpretation of channels as colors gives rise to graph coloring problems, it clearly stands in the tradition of the For Color Problem, hich itself rose from a practical problem ith restricted resorces. Distance Coloring, the assignment of colors to the vertices of a graph in sch a manner that vertices ithin a certain distance to each other receive different colors is one of the many variations of the Channel Assignment Problem and has some very exciting, and sometimes even srprising, implications in the theory of graphs that may go far beyond 3
Chapter 1 Introdction the original coloring problem. In this thesis e ill investigate some of these implications ith a strong focs on the algorithmic approachability. In terms of coloring problems this means the concepts of chordality and perfection hich are important parts of the fondation of algorithmic graph theory. This ill lead s, on varios occasions, aay from the original coloring problem and deeper into the strctral theory of graphs, bt eventally those detors ill prove fritfl. 4
Chapter 2 Definitions and Graph Coloring In this chapter e state some basic definitions in graph theory that are involved in this thesis. The concepts of coloring the vertices and edges of a graph are introdced alongside ith their generalizations and concepts of perfection in this context. Frthermore e give illstrating examples and prove some basic reslts. 2.1 Basic Terminology and Definitions Definition 2.1 (Graph). A graph G = (V,E) consists of to sets: the vertex set and the edge set, denoted by V(G) and E(G), or V and E hen no ambigities arise, the elements of V are called vertices and the elements of E, hich are 2-element sbsets of V, are called edges respectively. Each edge consists of to different vertices called endpoints. An edge e ith endpoints and v is denoted by e = v. The cardinality of the vertex setv is called the order ofg, denoted by V ; the cardinality of the edge set E is denoted by E and called the size of G. The graph hich has only one vertex and no edges is called the trivial graph. The graph G is finite if and only if V is a finite nmber. Definition 2.2 (Simple Graph). An ndirected graph is one in hich edges have no orientation. A loop is an edge connected at both ends to the same vertex. Mltiple edges are to, three or more edges connecting the same pair of vertices. A simple graph is an ndirected graph that has no loops and no mltiple edges. In this thesis, e only consider finite and simple graphs. Definition 2.3 (Neighborhood). To vertices and v are said to be adjacent in G if v E(G). Adjacent vertices are neighbors of each other. The set of all neighbors of any vertex in some GraphGis denoted by N G (), or N() hen no ambigities arise. The closed neighborhood of is given by N[] = N() {}. To edges e and e are said to be adjacent if they share a common endpoint and the edge neighborhood of e is denoted by N(e) = {e E(G) e e }. A vertex v is incident to an edge e - and vice versa - if v is an endpoint of e. 5
Chapter 2 Definitions and Graph Coloring Definition 2.4 (Degree). The degree of a vertex v in a graph G is the nmber of edges incident to v, denoted by deg G (v), or deg(v) hen no ambigities arise. If G is a simple graph it holds deg G (v) = N(v). The maximm degree of a graph G is given by (G) = max v V(G) deg G (v) and the minimm degree of G is given by δ(g) = min v V(G) deg G (v). Definition 2.5 (Paths and Cycles). Let G be a graph. For any,v V(G), a -v alk of G is a finite alternating seqence of vertices and edges of G hich begins ith vertex and ends ith vertex v, sch that each edge s endpoints are the preceeding and folloing vertices in the seqence. A alk is said to be closed if the seqence starts and ends at the same vertex. A -v path is a -v alk ith no repeated edges and vertices, denoted by P,v. The length of the path, denoted by P,v, is defined to be the nmber of edges in the seqence. A path is called an empty path if its length is zero, i.e., the seqence has one vertex and no edges. A cycle is a closed alk ith no repetitions of vertices and edges alloed, other than the repetition of the starting and ending vertex. A cycle of order, or length, n is denoted by C n, n is a positive integer and n 3 for any cycle. The girth girth(g) of a graph is the length of a longest cycle in G. Definition 2.6 (Distance). The distance beteen vertices and v in graph G, denoted by dist G (,v), is the length of a shortest path beteen and v. If the graph G is clear from the context, e denote the distance beteen vertices and v by dist(,v). Since only simple graphs are considered in this thesis an edge is niqely determined by its end vertices. Therefore e can se a seqence of vertices to represent a alk, path or cycle. In Figre 2.1 e find that the paths P 1 = a,b,c, P 2 = a,b,e,d,c, P 3 = a,f,e,d,c and P 4 = a,f,e,b,c are all possible a-c paths. Then dist(a,c) = min{p 1,P 2,P 3,P 4 } = 2. f e d a b c Figre 2.1: An example for paths and distance Definition 2.7 (k-neighborhood). For some vertex v V(G) the k-neighborhood of v is the set N k (v) = { V(G)\v dist G (v,) k}. The closed k-neighborhood is given by N k [v] = {v} N k (v). 6
2.1 Basic Terminology and Definitions Definition 2.8 (Connectivity). In a graph G, to vertices and v are called connected if G contains a path from to v. Otherise, they are called disconnected. A graph is said to be connected if every pair of vertices in the graph is connected. Connectivity defines an eqivalence relation on the set of vertices V, the eqivalence classes are called connectivity components or jst components of G. Definition 2.9 (Complete Graph). A complete graph K n, here n is a positive integer, is a graph of order n sch that every pair of distinct vertices in K n are connected by a niqe edge. The trivial graph is K 1. Definition 2.10 (Sbgraph). A sbgraph of a graphg = (V,E) is a graphh = (V H,E H ) ith V H V and E H E, denoted by H G. If H is a sbgraph of G, then G is a spergraph of H. Frthermore, the sbgraph H is called an indced sbgraph of G on V H if v E implies v E H for any,v V H. Let U V, the indced sbgraph of G on the vertex set U is denoted by G[U]. For some set W E(G) of edges the edge indced sbgraph G[W] is the graph indced by the vertex set e W. Definition 2.11 (F-free). We say that G contains a graph F, if F is isomorphic to an indced sbgraph of G. A graph is F-free if it does not contain F, and for a family of graphs F, G is F-free if G is F-free for every F F. Definition 2.12 (Cliqe and Stable Set). The set C V is called a cliqe if and only if the indced sbgraph G[C] is a complete graph. We denote the maximm cardinality of sch a sbset C of V by ω(g). The set I V is called independent or stable if and only if the indced sbgraph G[I] does not contain any edges. We denote the maximm cardinality of sch a sbset I of V by α(g). Consider the graph G in Figre 2.2, the graphs H 1 and H 2 are both sbgraphs of G. H 1 contains all edges that exist in G beteen the vertices U = { 1,v 1,v 2,v 3,v 4 }, therefore it holds G[U] = H 1 and H 1 is an indced sbgraph of G. The graph H 2 on the other hand lacks the edge v 1 v 4 and therefore is not an indced sbgraph of G. The graph C too is an indced sbgraph of G and in addition C = K 3, so C is a cliqe in G, i.e. C is a maximm cliqe of G and ω(g) = 3. At lasti is an indced sbgraph ofg, in particlar one that does not contain edges hich makes it a stable set of vertices. Althogh I is maximal in the manner that no other vertex of G cold be inclded ithot prodcing an edge, it is no maximm stable set. We have α(g) = 4 and the only maximm stable set of G is given by { 1, 2, 3, 4 }. Definition 2.13 (Basic Operations). For some sbset of vertices S V(G) of G. G S denotes the indced sbgraph G[V(G)\ S] of G, hich reslts from G by deleting S. If 7
Chapter 2 Definitions and Graph Coloring G v 1 1 v 4 H 1 v 1 1 v 4 H 2 v 1 1 v 4 2 4 v 2 v 3 v 2 v 3 v 2 v 3 3 I 1 C v 4 4 2 v 3 v 3 Figre 2.2: An example for indced sbgraphs, cliqes and stable sets. S = 1 ith S = {v} e rite G v. For deleting edges e have to differentiate beteen to possibilities, deletion and sbtraction. Let W E(G), deleting the edges means the removal of W from G, denoted by G\W, hich reslts in the graph(v(g),e(g)\w), sbtraction, denoted by G W, means the indced sbgraph obtained by the deletion of the endpoints of all edges in W, hich reslts in the graph G ( e W e). Definition 2.14 (Line Graph). The line graph L(G) of a graph G is the graph defined on the edge set E(G) (each vertex in L(G) represents a niqe edge in G) sch that to vertices in L(G) are defined to be adjacent if their corresponding edges in G share a common endpoint. The distance beteen to edges in G is defined to be the distance beteen their corresponding vertices in L(G). For to edges e,e E(G) e rite dist G (e,e ) = dist L(G) (e,e ). Definition 2.15 (Complement). The complement of a graph G = (V,E) is given by G = ( V,E ), here E denotes the edges ( ( V 2) \ E and V 2) is the set of all sbsets of V ith cardinality 2. Definition 2.16 (Tree). A tree T is a simple connected graph that has no cycles. Any vertex in a tree ith degree one is called a leaf. Definition 2.17 (Star). A star is a tree ith exactly n 1 leaves, here n is the order of the tree. 8
2.2 Vertex Coloring A doblestar is a tree ith exactly n 2 leaves, here n is the order of the tree. If the degrees of the to non-leaf vertices of a doble star are x and y, here deg(x) deg(y), e denote sch a doble star by D x,y. Definition 2.18 (Chordal Graph). Let C = (v 1,...,v n ) be a cycle, the graph indced by the vertices of C is denoted by G[C]. If G[C] = C the cycle is called chordless, otherise there exists some edge v i v j in G ith j i±1(mod n), this edge is called a chord. If G does not contain a chordless cycle of length 3 it is called chordal. Definition 2.19 (Planar Graph). A planar graph is a graph that can be embedded in the plane, i.e., it can be dran on the plane in sch a ay that its edges intersect only at their endpoints. A plane graph is a draing of a planar graph on the plane sch that no edges cross each other. A plane graph divides the plane into areas srronded by edges, called faces (or regions). The area otside the plane graph is the nbonded face of the graph. 2.2 Vertex Coloring Definition 2.20 (Proper Vertex Coloring). For a graph G, a vertex coloring is a mapping F of the vertices of G to some set of colors, hich sally are represented by integers: F : V(G) {1,...,c}. If F ses exactly c different colors, e call F a c-coloring of G. The color classes F 1 (i) = F i give a partition of the vertex set of G. A proper vertex coloring is a vertex coloring F ith xy E(G) F(x) F(y). In this thesis a proper vertex coloring ell simply be referred to as a coloring. The chromatic nmber χ(g) of G is the smallest integer c for hich a proper c-coloring of G exists. The neighborhood constraint of a proper vertex coloring can be rephrased in terms of the distance beteen vertices. dist G (x,y) 1 F(x) F(y) A natral generalization of proper vertex coloring seems to be the extension of the distance in hich to vertices are not alloed to have the same color. The idea of coloring objects ith some constraint to their distance as first introdced by Eggleton, Erdős and Skilton in their ork on the chromatic nmber of distance graphs ([EES85]) and then as extended to general graphs by Sharp ([Sha07]), ho calls colorings of this type distance colorings. For better compatibility in terms of terminology ith the main chapters of this thesis e ill refer to sch colorings as strong colorings. 9
Chapter 2 Definitions and Graph Coloring Definition 2.21 (k-strong Vertex Coloring). For a graph G and some positive integer k N, a k-strong vertex coloring or simply k-strong coloring is a vertex coloring F : V(G) {1,...,c} ith dist G (x,y) k F(x) F(y) for all x,y V(G), x y. The k-strong chromatic nmber χ k (G) of G is the smallest integer c for hich a k-strong c-coloring of G exists. While for all graphs G the simple connection beteen the cliqe nmber of G and its chromatic nmber ω(g) χ(g) holds, e can derive a similar relationship for all k by generalizing the concept of cliqes in the same manner as e did ith colorings. Definition 2.22 (k-strong Cliqe). The set C V(G) is called a k-strong cliqe for some positive integer k N if and only if for each pair x,y C it holds dist G (x,y) k. The k-strong cliqe nmber ω k (G) is the maximm cardinality of sch a sbset C of V(G). Definition 2.23 (k-strong Neighborhood). The set N k (v) = { V(G)\{v} dist G (v,) k} is called the k-strong neighborhood or k-neighborhood of v. There are some basic observations hat can be made for those parameters and allk N. i) ω k 1 (G) ω k (G) ii) χ k 1 (G) χ k (G) iii) ω k (G) χ k (G) The case k = 1 clearly resembles the standard k-coloring problem. A ell knon pper bond to the chromatic nmber is given in dependency of the maximm degree (G). A generalized version for k-strong colorings ill be investigated in the next chapter. Theorem 2.24. For any graph G it holds χ 1 (G) (G)+1. A special and very important class of graphs is the class of so called perfect graphs hich as introdced first by Clade Berge in 1963. Not only are those graphs very important de to their strctral properties, bt also de to some comptational reslts to hich e ill come back later. The definition of perfection hich e give in this thesis is not the same as the one that as introdced by Berge, althogh it is eqivalent de to the so called eak perfect graph theorem. 10
2.2 Vertex Coloring Definition 2.25 (Perfection). A graph G is called perfect if for all indced sbgraphs H G it holds ω(h) = χ(h). In terms of k-strong colorings e get the generalized definition. Definition 2.26 (k-strong Perfection). A graph G is called k-strong perfect or k-perfect if for all indced sbgraphs H G holds ω k (H) = χ k (H). For the case k = 1, the general perfection, several very important characterizations are knon, especially the so called eak and the strong perfect graph theorem. Theorem 2.27 (Weak Perfect Graph Theorem, Lovász. 1972 [Lov87]). A graph G is 1-perfect if and only if G is 1-perfect. Theorem 2.28 (Strong Perfect Graph Theorem, Chdnovsky, Robertson, Seymor, Thomas 2002 [CRST06]). A graph G is 1-perfect if and only if neither G nor G contains an indced circle of odd length 5. What exactly does a proper vertex coloring of G correspond to in the complement? For some proper coloring F a color class F i indces a stable set in G, therefore its indced graph in G is complete. We cover the vertices of G ith cliqes and get the folloing. Definition 2.29 (Cliqe Cover). For a graph G, a cliqe cover is a vertex c-coloring C ith x,y C i xy E(G), here C i is a color class of C. The cliqe cover nmber χ(g) of G is the smallest integer c for hich a c-cliqe cover of G exists. Again e can generalize the concept of cliqe covers in terms of the distance beteen vertices and get a strong version. Definition 2.30 (k-strong Cliqe Cover). For a graph G and some positive integer k N, a k-strong cliqe cover is a vertex c-coloring C ith x,y C i dist G (x,y) k The cliqe cover nmberχ k (G) of G is the smallest integer c for hich ak-strong c-cliqe cover of G exists. Definition 2.31 (k-strong Stable Set). The set I V(G) is called a k-strong stable set for some positive integer k N if and only if for each pair x,y I, x y, it holds dist G (x,y) > k. The k-strong stability nmber α k (G) is the maximm cardinality of sch a sbset I of V(G). 11
Chapter 2 Definitions and Graph Coloring Some basic observations similar to those regarding cliqes and colorings can be made for all k N. The difference beteen the to concepts is that, hile a k-strong cliqe gets bigger and bigger ith increasing k and therefore more colors are needed, a k-strong stable set gets smaller ith increasing k and therefore feer k-strong cliqes are reqired to cover the hole graph. i) α k (G) α k 1 (G) ii) χ k (G) χ k 1 (G) iii) α k (G) χ k (G) A ell knon reslt regarding a family of perfect graphs hich ill be stdied very intense in this thesis is the folloing. Theorem 2.32. Every chordal graph is perfect. 2.3 Edge Coloring The coloring of edges is somehat of a special case of vertex coloring. The standard definition of a proper edge coloring indces a coloring of the vertices of the line graph. Definition 2.33 (Proper Edge Coloring). For a graph G, an edge coloring is a mapping F of the edges of G to some set of colors: F : E(G) {1,...,c}. A proper edge coloring is an edge coloring F ith e e F(e) F(e ) for all e,e E(G). The chromatic index χ (G) is the smallest integer c for hich a proper c-edge coloring of G exists. A pair of edges e and e that share a common endpoint and therefore flfill e e correspond to to adjacent vertices in the line graph. Therefore a proper edge coloring of some graph G corresponds to a proper vertex coloring of L(G), χ (G) = χ(l(g)). With that e can give a definition of the generalized k-strong edge coloring by sing the indced distance in the line graph. Definition 2.34 (k-strong Edge Coloring). For a graph G and some positive integer k N, a k-strong edge coloring is an edge coloring F : E(G) {1,...,c} ith dist L(G) (e,e ) k F(e) F(e ) for all e,e E(G), e e. The k-strong chromatic index χ k (G) of G is the smallest nmber c for hich a k-strong edge c-coloring of G exists. 12
2.3 Edge Coloring A proper edge coloring partitions the set of edges into sbsets here no to edges share a common endpoint. Sch a set of edges is called a matching. Definition 2.35 (Matching). The set M E(G) is called a matching if and only if for all pairs of edges e,e M it holds e e. We denote the maximm cardinality of sch a sbset M of E by ν(g). Definition 2.36 (k-strong Matching). The set M E(G) is called a k-strong matching for some positive integer k N if and only if for all pairs of edges e,e M it holds dist L(G) (e,e ) > k. We denote the maximm cardinality of sch a sbset M of E by ν k (G). Obviosly a star represents a strctre hich is somehat similar to a cliqe in the ay that no more than one edge of a star can be part of the same matching. It is de to this property that those strctres are called anti-matchings. Definition 2.37 (k-strong Anti-Matching). The set A E(G) is called a k-strong antimatching for some positive integer k N if and only if for all pairs of edges e,e A it holds dist L(G) (e,e ) k. We denote the maximm cardinality of sch a sbset A of E by am k (G). The relation beteen k-strong anti-matchings and k-strong cliqes becomes obvios by translating the definition of k-strong anti-matchings to the langage of line graphs. We get the folloing relations. i) am k (G) = ω k (L(G)) ii) am k (G) χ k (G) With another translation e obtain the last remaining dal concept in terms of taking the line graph of a graph: The anti-matching cover of a graph. Definition 2.38 (k-strong Anti-Matching Cover). For a graph G and some positive integer k N, a k-strong anti-matching cover is an edge coloring C ith e,e C i dist L(G) (e,e ) k. The k-strong anti-matching cover nmber amc k (G) is the smallest integer c for hich a k-strong c-anti-matching cover of G exists. Again e can derive the knon ineqality ν k (G) = α k (L(G)) χ k (L(G)) = amc k (G). In addition it is possible to sho some basic relation beteen k-strong anti-matchings and cliqes in the same graph. 13
Chapter 2 Definitions and Graph Coloring Lemma 2.39. Let k be a positive integer and A a k-strong anti-matching of a graph G. Then C = e Ae is a (k +1)-strong cliqe in G. Proof. Let A E(G) be a k-strong anti-matching in G, then it sffices to sho dist G (v,) k +1 for all v, C = e A e. For each pair of edges e,e A it holds dist L(G) (e,e ) k and ith that a shortest e-e -path in L(G) contains at most k edges and k 1 other vertices, corresponding to another k 1 edges in G. In general, any path of length l in the line graph, connecting to vertices a and b of the line graph corresponds to a path in the original graph connecting the to edges corresponding to the vertices a and b. This path, in G, connects the to endpoints of a and b that have the smallest distance to each other and ths is by 1 shorter than the path in the line graph. Therefore there exists a pair of verticesx,x ithx e andx e satisfyingdist G (x,x ) k 1. Frthermore for e = xy and e = x y e get dist G (y,y ) k +1. Lemma 2.40. Let k be a positive integer and C a k-strong cliqe of a graph G. Then E(G[C]) is a (k +1)-strong anti-matching in G. Proof. Let e,e E(G[C]) ith e = xy ande = x y, then it holds.l.o.g. dist G (x,x ) k and therefore there exists a path from x to x in G sing at most k edges. This path corresponds to a path from e to e in L(G) sing at most k vertices and therefore at most k +1 edges. Hence dist L(G) (e,e ) k +1. One cold conjectre the existence of some tighter bond, i.e. the correspondence of a k-strong cliqe to a k-strong anti-matching and the other ay arond. The folloing examples ill sho that generally no sch thing exists. Example 2.41. C 6 v 3 v 2 v 4 v 1 v 5 v 6 The edges in A = {{v 1,v 2 },{v 3,v 4 },{v 5,v 6 }} form a maximm 2-strong anti-matching in C 6, simltaneosly V(C 6 ) = e Ae forms a 3-strong cliqe. This is the exact reslt hich follos from Lemma 2.39 and in fact E(C 6 ) forms a 3-strong anti-matching. 14
2.3 Edge Coloring P 4 v 1 v 2 v 3 v 4 v 5 In a path of length 4 the to oter vertices are in distance exactly 4, hence P 4 is a 4-strong cliqe. The line graph again is a path, bt shortened by 1, therefore P 4 is a 3-strong anti-matching. The graph P 4 shos that e cannot improve Lemma 2.39. The other ay arond, a k-strong cliqe hich also is a (k +1)-strong anti-matching is hardly that obvios. W v 6 v 5 v 4 v 3 v 7 v 2 v 8 v 1 v 9 v 10 v 11 v 12 In the graph W all pairs of vertices x,y V(W 4 ) satisfy dist W4 (x,y) 3, hence W forms a 3-strong cliqe. On the other hand the edges v 1 v 2 and v 7 v 8 are ithin a distance of 4, therefore W forms a 4-strong anti-matching and hence Lemma 2.40 too cannot be improved. Several other extensions of the ordinary chromatic nmber have been proposed and stdied in the past and some of them, or better, the perspective proposed by them has some interesting impact on the relation beteen strong vertex and strong edge coloring. One of those concepts is the acyclic coloring (see [Bor79], [AMS96] or [HW15] for some details) hich is the minimm nmber of colors in a proper vertex coloring sch that the vertices of any to colors indce an acyclic graph. This concept has been transported to the realm of edge colorings, here an acyclic edge coloring is a proper edge coloring for hich the edges of any cycle se at least three different colors (see [ASZ01], [AZ02] and [VS16] for details). This particlar parameter seems to even have some important application in chemistry and might be interesting simply de to this fact (see [RRP15]). If e investigate the 2-strong vertex coloring or the 2-strong edge coloring, e soon 15
Chapter 2 Definitions and Graph Coloring stmble pon some very basic principals in terms of loer bonds. For a 2-strong vertex coloring the vertices of a path of length 2 may not have the same color, ths any cycle ith at least three vertices has at least three different colors assigned to its vertices. Similar observations can be made for 2-strong edge colorings and so it becomes clear that χ 2 (G) and χ 2(G) pose pper bonds on the acyclic chromatic nmber and the acyclic chromatic index. Another interesting concept of vertex colorings is the star coloring, hich is a stronger version of the acyclic coloring (see [Vin88], [AZ93] and [VJM15] for details). Here no it is reqired that the vertices of any to color classes of a star coloring indce a so called star forest. Again e can dra some relations to the 2-strong coloring, hich poses as an pper bond on the star chromatic nmber as ell, since, as e ill see next, any to color classes of a proper 2-strong coloring indce a 2-strong matching, hich is a special case of a star forest. Lemma 2.42. Let G be a graph and F a 2-strong coloring of G, then the vertices of any to color classes F i and F j of F indce a 2-strong matching of G. Proof. Sppose F i and F j do not indce a 2-strong matching of G, then there mst exist a path of length 2 in G[F i F j ], say P. Let P = xyz, then no to of the three vertices may have the same color, ths e need at least three colors for a proper 2-strong coloring of G[F i F j ], a contradiction. So, if e assign the tple of the to different colors nder the given 2-strong coloring F as a color to every edge, e obtain a proper 2-strong edge coloring of the same graph. Corollary 2.43. Let G be a graph, the folloing ineqality holds: ( ) χ 2 (G) χ2 (G). 2 A somehat similar, bt a bit eaker, reslt is tre if e sap vertex and edge coloring and assign any pair of to colors of edges incident to a vertex in G as its color. Note that in this case the coloring of the vertex is not clear, since a vertex may be incident to more than to edges. Bt since the folloing holds for any to color classes, any pair of sch colors may be chosen. Lemma 2.44. Let G be a graph and F a 2 strong edge coloring of G, then the vertices of degree 2 in the indced sbgraph of the vertices belonging to the edges of any to color classes F i and F j of F form a stable set. 16
2.3 Edge Coloring Proof. Sppose in the graph indced by the vertices of F i and F j, call it G ij, are to vertices x and y of degree 2 ith dist G (x,y) = 1. If they are adjacent, they are adjacent in G ij too and ith both of them having a degree of to e can find a path of length 3 in G ij. No there are at least three colors necessary to color the edges of a path of length 3 in a 2-strong edge coloring, ths this is a contradiction to F being a proper 2-strong edge coloring. Corollary 2.45. Let G be a graph, the folloing ineqality holds: ( ) χ χ(g) 2 (G). 2 The folloing example shos that in general the first bond is tight. For the second bond e give an example ith a big gap, that shos that the actal error made by the procedre proposed above can be mch smaller. We are confident that those bonds can easily be extended to general k-strong colorings. Example 2.46. (1,2) 2 (2,3) 3 2 (1,2) 1 (1,3) 1 Figre 2.3: A 2-strong edge coloring constrcted from a 2-strong coloring. For this graph G e have χ 2 (G) = 3 and χ 2(G) = 3 = ( 3 2) and ths the bond of Corollary 2.45 is tight for G. Example 2.47. (1) (1) 1 (1,2) 2 3 (3,4) 4 1 (1,2) 2 (2) (2) Figre 2.4: A ordinary vertex coloring constrcted from a 2-strong edge coloring. 17
Chapter 2 Definitions and Graph Coloring For this graph G e have χ 2 (G) = 4 and χ(g) = 2, bt or procedre assigns 4 different colors, hile the bond by Corollary 2.45 gives s a vale of ( 4 2) = 6. This example also shos that the procedre sed in the proof of Lemma 2.44 by far is not optimal and hy e did not obtain a bond on the 2-strong chromatic nmber. Bt it sggests that e cold obtain sch a bond from the same strctre, if e changed it a little, since changing one of the to (1,2) labels to one of its alternatives (resp. (1,3) or (1,4)) old reslt in a proper 2-strong vertex coloring in 5 colors here jst 4 are needed. As e ere able to transfer the concepts of coloring the vertices of a graph to the edges hile preserving cliqe-like strctres and stable sets, e can give an analog definition of perfection in terms of edge coloring. The concept of perfection of graphs regarding strong edge colorings as first introdced by Chen and Chang ([CC14]). Definition 2.48 (k-strong am-perfection). A graph G is called k-strong am-perfect if for all edge indced sbgraphs H G holds am k (H) = χ k(h). 18
Poers of Graphs Chapter 3 In this chapter e ill give a brief smmary of reslts regarding pper bonds on the chromatic nmber of graph poers, e then ill have a qick look into some special graph classes hich are closed nder the taking of poers and some share the attribte of being chordal, hich leads s to the characterization of chordal graphs closed nder the taking of poers by Laskar and Shier. We then proceed to general poers of chordal graphs, here e ill see some very exciting reslts in terms of creating cycles throgh taking the poer of graphs. This leads to a characterization of graphs hose sqares become chordal and even a first attempt to find graphs hose sqares are perfect. Distance colorings, or in or notation strong colorings, ere also stdied nder another name. Instead of jst defining some specific coloring ith restrictions in the distance beteen vertices one cold add more edges to the graph connecting those vertices ithin a given distance to each other. This prodces a ne graph hose reglar chromatic nmber is the same as the k-strong chromatic nmber of the original graph. This is knon as taking the k-th poer of a graph and has been stdied very intense not only in terms of colorings. Definition 3.1 (Poer of Graphs). The k-th poerg k of a graph G = (V,E) is given by the vertex set V ( G k) = V and the edge set E ( G k) = {v v, V, dist G (v,) k}. As mentioned above the folloing relations hold for the k-th poer of a graph G: i) ω k (G) = ω ( G k) ii) α k (G) = α ( G k) iii) χ k (G) = χ ( G k) iv) χ k (G) = χ ( G k) There have been some attempts on general bonds of the chromatic nmber of graph poers. Especially the case k = 2, the so called sqare of a graph, has experienced a lot of attention. 19
Chapter 3 Poers of Graphs 3.1 Upper Bonds on the Chromatic Nmber of Graph Poers We ill begin here ith a very basic observation in terms of the maximm degree and then explore more sophisticated bonds that make se of the girth, or of properties of special graph classes. A generalized version of Theorem 2.24 can be obtained via a simple indction. Lemma 3.2. For any graph G and positive integer k N it holds χ k (G) (G) k +1. Proof. With χ k (G) = χ ( G k) e can make se of Theorem 2.24 and redce the proof to shoing ( G k) (G) k. No let G be a graph ith maximm degree and v V(G) ith deg G (v) =. By definition e have deg G (x) for all x N i (v) ith i {1,...,k}. We get deg G k(v) (G)+ (G)( (G) 1)+ (G)( (G) 1) 2 + + (G)( (G) 1) k 1 (G) k, by proving the second ineqality by indction over k. The natral pper bond is even better if not compressed into (G) k. For graphs ith (G) > 1 the folloing transformation can be applied. (G) k i=1 ( (G) 1) i 1 = (G) (G)k+1 1 (G) Theorem 3.3. For any graph G ith (G) > 1 and any positive integer k N it holds χ k (G) (G) (G)k+1 1 (G) +1. Alon and Mohar ere able to improve this for k = 2 on graphs ith bonded girth. The cases ith 3 girth(g) 6 and girth(g) 7 are considered separate. Theorem 3.4 (Alon, Mohar 2002 [AM02]). Let G be a graph ith 3 girth(g) 6, then it holds χ 2 (G) (1+o(1)) (G) 2. 20
3.1 Upper Bonds on the Chromatic Nmber of Graph Poers Theorem 3.5 (Alon, Mohar 2002 [AM02]). Let G be a graph ith girth(g) 7, then it holds ( ) (G) 2 χ 2 (G) Θ. log (G) To be precise they ere able to sho a mch more interesting fact for this pper bond of the chromatic nmber of sqared graphs. With f 2 (,g) they defined some fnction for the maximm possible vale for χ 2 (G) for all graphs ith maximm degree and girth g and proved the folloing reslt by sing the probabilistic method. Theorem 3.6 (Alon, Mohar 2002 [AM02]). i) There exists a fnction ǫ( ) that tends to 0 as tends to infinity sch that for all g 6 (1 ǫ( )) 2 f 2 (,g) 2 +1. ii) There are absolte positive constants c 1, c 2 sch that for every 2 and every g 7 c 1 2 log f 2(,g) c 2 2 log. They ere able to generalize those reslts to higher poers by extending their random constrctions sed in the proof of Theorem 3.6, for k 3 they obtained the folloing bonds for the generalized pper bond fnction f k (,g). Theorem 3.7 (Alon, Mohar 2002 [AM02]). There exists an absolte constant c > 0 sch that for all integers k 1, 2 and g 3k +1 f k (,g) c k k log. For every integer k 1 there exists a positive nmber b k sch that for every 2 and g 3 f k (,g) b k k log. Coloring the vertices of planar graphs has a long tradition in chromatic graph theory. In 1977 Wegner (see [Weg77]) first investigated the chromatic nmber of sqared planar graphs. He as able to prove the simple bond χ 2 (G) 8 for every planar graph G ith maximm degree (G) = 3. He also conjectred that this bond cold be improved to 21
Chapter 3 Poers of Graphs 7, hich as confirmed by Thomassen (see [Tho01]) in 2001. Wegner gave a conjectre for bonds of the chromatic nmber of sqared planar graphs ith higher maximm degrees too, along ith examples that prove these bonds to be tight if his conjectre is correct. Conjectre 3.8 (Wegner. 1977 [Weg77]). Let G be a planar graph. Then { (G)+5, if 4 (G) 7, χ 2 (G) +1, if (G) 8. 3 (G) 2 While there ere many attempts to prove Wegner s conjectre it remains open. So far the best bonds regarding this specific problem ere obtained by Molloy and Salavatipor (see [MS05]) in 2005, they gave both a general and an asymptotic bond. Theorem 3.9 (Molloy, Salavatipor. 2005 [MS05]). Let G be a planar graph. Then 5 χ 2 (G) 3 (G) +78. Theorem 3.10 (Molloy, Salavatipor. 2005 [MS05]). Let G be a planar graph ith (G) 241. Then 5 χ 2 (G) 3 (G) +25. By forbidding the complete graph on 4 vertices as a minor - note that the K 5 already is forbidden in planar graphs and ith K 3,3 also containing the K 4 as a minor those graphs are a tre sbset of planar graphs - Lih, Wang and Zh (see [LWZ03]) ere able to frther improve these bonds for this sbclass. Theorem 3.11 (Lih, Wang, Zh. 2003 [LWZ03]). Let G be a K 4 -minor free graph. Then { (G)+3, if 2 (G) 3, χ 2 (G) = +1, if (G) 4. 3 (G) 2 3.2 Graph Classes Closed Under Taking Poers Another approach to the problem of chromatic nmbers of graph poers is the qestion hich families of graphs are closed nder the taking of poers. 22
3.2 Graph Classes Closed Under Taking Poers Definition 3.12. Let F be a family of graphs, F is called closed nder taking poers if G k F implies G k+1 F for all k N. The connection beteen sch a family and the chromatic nmber of graph poers lies in the the fact that a great nmber of sch families, hich are knon to be closed nder taking poers, that are sbclasses of chordal graphs and therefore perfect. The first knon reslt in this type of stdies in graph poers as obtained by Lbi (see [Lb82]). His family is one of very important chordal graphs, the so called strongly chordal graphs hich can be characterized in terms of indced sbgraphs. Those sbgraphs are a very important special case of a graph class hich e ill stdy in the next section. Definition 3.13 (Sn). A sn is a graph S n = (U W,E) ith U = { 1,..., n } and W = { 1,..., n } sch that W indces a complete graph and U is a stable set and frthermore i v j E if and only if j = i or j = i+1(mod n). 1 2 1 5 5 2 4 3 3 4 Figre 3.1: The sn ith n = 5. Theorem 3.14 (Farber. 1983 [Far83]). A chordal graph is strongly chordal if and only if it does not contain a sn. Note that Farber called those forbidden sbgraphs not sns bt trampolines, later on the term sn became more poplar. Theorem 3.15 (Lbi. 1982 [Lb82]). If G k is a strongly chordal graph, so is G k+1 for all k N. There are many knon characterizations of strongly chordal graphs and to of them are in terms of so called totally balanced matrices. Totally-balanced matrices ere sed by Lovász (see [Lov14]) and are those matrices not containing the incidence matrix of 23
Chapter 3 Poers of Graphs a cycle of length at least three. Ths a hypergraph is totally balanced if and only if its incidence matrix is totally balanced. Hence a graph G is totally balanced if and only if G is a forest. Althogh strongly chordal graphs in general may contain cycles there is a strong relation beteen them and totally balanced matrices. Definition 3.16 (Neighborhood Matrix). The neighborhood matrix M(G) of a graph G on n vertices v 1,...,v n is the n n matrix ith entry (i,j) eqals 1 if and only if v i N[v j ] and 0 otherise. Theorem 3.17 (Lbi. 1982 [Lb82]). A graph G is strongly chordal if and only if M(G) is totally balanced. Definition 3.18 (Cliqe Matrix). The cliqe matrix C(G) of a graph G on n vertices v 1,...,v n ith maximal Cliqes C 1,...,C q is the q n matrix ith entry (i,j) eqals 1 if and only if v j C i and 0 otherise. Theorem 3.19 (Lbi. 1982 [Lb82]). A graph G is strongly chordal if and only if C(G) is totally balanced. The strctre of strongly chordal graphs allos a lot of problems, that remain hard to solve on general chordal graphs, to be solved efficiently. In chapter 5 e ill revisit this class of graphs. Definition 3.20 (Intersection Graph). The intersection graph of a family I of sets is the graph ho has a vertex for every set in I and to distinct vertices are adjacent if and only if their corresponding sets intersect. The concept of intersection graphs plays an important role in graph theory, especially in the stdy of different classes of chordal graphs. Given certain strctres to the sets in I the corresponding intersection graph often holds very strong strctral properties. One of the most excessively stdied family of intersection graphs are the interval graphs i.e. the intersection graphs of intervals on the real line. The closre of interval graphs nder the taking of poers as first shon by Raychadhri (see [Ray87]) bt Chen and Chang (see [CC01]) gave a mch simpler proof in terms of a characterization by Ramalingam and Rangan. Theorem 3.21 (Ramalingam, Rangan. 1988 [RR88]). A graph G is an interval graph if and only if it has an interval ordering, hich is an ordering of V(G) into [v 1,...,v n ] sch that i < l < j and v i v j E(G) v l v j E(G). Theorem 3.22 (Raychadhri. 1987 [Ray87]). If G k is an interval graph, so is G k+1 for all k N. 24
3.2 Graph Classes Closed Under Taking Poers Proof. Let G k be an interval graph and σ = [v 1,...,v n ] an interval ordering of G k. We ant to sho that σ is an interval ordering for G k+1 as ell. No sppose i < l < j and v i v j E ( G k+1), hich implies dist G (v i,v j ) k + 1. If dist G (v i,v j ) k, then e get v i v j E ( G k) and ith σ being an interval ordering of G k v l v j E ( G k) E ( G k+1). No sppose dist G (v i,v j ) = k + 1. Let P be a shortest v i,v j -path in G and let v a be the vertex adjacent to v j on P. Then, dist G (v i,v a ) = k and dist G (v a,v j ) = 1 hence v i v a and v a,v j are edges in G k. If i < l < a, then v l v a exists in G k by σ and so dist G (v j,v j ) dist G (v l,v a )+dist G (v a,v j ) k+1 holds and v j is adjacent to v l in G k+1. If a < l < j, then v l v j exists in G k and therefore in G k+1, in any case v l and v j are adjacent and σ is an interval ordering of G k+1, hich, by Theorem 3.22, makes G k+1 an interval graph. Remark 3.23. Strongly chordal graphs and interval graphs are chordal. An even larger class of graphs closed nder the taking of poers and inclding interval graphs is given by the so called cocomparability graphs. Jst like interval graphs e can give a special ordering σ of the vertices of sch a graph in order to not only characterize them, bt to prove the closre. Definition 3.24 (Comparability Graph). A comparability graph is the nderlying graph of an acyclic digraph, hich can be vieed as a partially ordered set (poset). I.e. a graph G is a comparability graph if and only if it has a transitive ordering σ = [v 1,...,v n ] of V(G) sch that i < l < j and v i v j E(G) v l v j E(G). Definition 3.25 (Cocomparability Graph). A cocomparability graph is the complement of a comparability graph, i.e. it has a cocomparability ordering σ = [v 1,...,v n ] of V(G) sch that i < l < j and v i v j E(G) v i v l E(G) or v l v j E(G). Again, hile originally proven by Floto in 1995 (see [Flo95]) Chen and Chang (see [CC01] gave a mch simpler and more elegant proof. Theorem 3.26 (Floto. 1995 [Flo95]). If G k is a cocomparability graph, so is G k+1 for all k N. Proof. Similar to the proof for interval graphs e take some cocomparability graph G k for some k N and a corresponding cocomparability ordering σ = [v 1,...,v n ] and sho that σ is a cocomparability ordering of G k+1 as ell. Sppose i < l < j and v i v j E ( G k+1), hence dist G (v i,v j ) k +1. If dist G (v i,v j ) k 25
Chapter 3 Poers of Graphs holds either v i v l orv l v j exist ing k ith σ being a cocomparability ordering and therefore at least one of those to edges exists in G k+1. So e sppose dist G (v i,v j ) = k + 1 and chose some vertex v a on a shortest v i,v j -path in G ith dist G (v i,v a ) = k and dist G (v a,v j ) = 1. With that v a is adjacent to v i and v j in G k. If i < l < a, then either dist G (v i,v l ) k or dist G (v l,v j ) k is implied by the existence of either one of the corresponding edges in G k. If v i v l exists in G k it does so in G k+1 too and e are done, so sppose v l,v j E ( G k). We get dist G (v l,v j ) dist G (v l,v a )+dist G (v a,v j ) k +1 and so the edge v l v j exists in G k+1. Ths e reach the case a < l < j and either the edge v a v l or v l v j exists in G k, hich implies the distance conditions dist G (v a,v l ) k or dist G (v l,v j ) k. With dist G (v l,v j ) k the edge v l v j exists in both G k and G k+1. And for dist G (v a,v l ) k e get dist G (v l,v j ) dist G (v a,v l )+dist G (v a,v j ) k+1 and therefore the existence of v l v j in G k+1. Hence σ is a cocomparability order of G k+1 and therefore G k+1 is a cocomparability graph. Remark 3.27. Cocomparability graphs are perfect. 3.2.1 Chordal Graphs In general chordal graphs are not closed nder the taking of poers. Nevertheless it is possible to sho a very similar property. In 1980 Laskar and Shier (see [LS80]) shoed that hile G 3 and G 5 are chordal if G itself is chordal, bt in general the same does not hold for G 2. We have already seen an example of sch a graph. The sn S 5 in Figre 3.1. Bt sch graphs do not necessarily need to be sns. Chordal graphs that look a lot like sns, bt are none, have this property too. 1 2 1 5 5 2 4 3 3 4 Figre 3.2: Another example of a chordal graph hose sqare is not chordal. This led Laskar and Shier to the conjectre that every odd poer of a chordal graph is chordal itself. Dchet (see [Dc84]) proved an even stronger reslt hich led to a 26
3.2 Graph Classes Closed Under Taking Poers nmber of so called Dchet-type reslts on graph poers. Dchet s proof makes se of very basic tools sch as alks. For chordal graphs the folloing lemma ill be very sefl. Lemma 3.28 (Dchet. 1984 [Dc84]). Every closed alk v 1,...,v n,v 1 ith n 2 contains a chord v i v i+1 or a repetition v i = v i+1. In order to prove Dchet s Theorem e need an additional definition. Definition 3.29 (Graph Modlo Sbset). Let G be a graph and V 1,...,V m V(G) sbsets of V(G). The graph G(V 1,...,V m ) has the vertex set {V 1,...,V m } and V i and V j are adjacent if and only if ( v V i N(v) ) V j. Lemma 3.30 (Dchet. 1984 [Dc84]). Let G be a chordal graph, if V 1,...,V m are connected sbsets of V(G), G(V 1,...,V m ) is also chordal. Proof. Let C = C 1...C q C 1 be a cycle ith length q 4 in G(V 1,...,V m ). We call a closed alk W = 1,..., p in G a C-alk if and only if there is a decomposition of W into q sbalks W i here W i is a nonempty alk in G[C i ]. W i is called the i-th component of the C-decomposition W 1,...,W q. No e consider a C-alk W ith minimal length p, ith vertices v 1,...,v p and W 1,...,W q thec-decomposition of W. Lemma 3.28 implies the existence of to vertices v t and v t+2 sch that v t = v t+2 or v t is adjacent to v t+2. The minimality of W implies frthermore that v t and v t+2 are in different components, otherise W cold be shortened by ctting ot the vertex v t+1 either by sing the edge v t v t+2 or by jst not sing the edge v t v t+1 in W. Let W α and W β be those different components ith α β 1, (α,β) (q,1) and (α,β) (1,q). Therefore W α and W β are linked in G(V 1,...,V m ), hich reslts in a chord in C, ths G(V 1,...,V m ) is chordal. Theorem 3.31 (Dchet s Theorem, Dchet. 1984 [Dc84]). Let G be a graph and k N. If G k is chordal, so is G k+2. Proof. LetV(G) = {v 1,...,v n }, e define V i := N k (v i ) {v i } for all i {1,...,n}. Then G k (V 1,...,V n ) is isomorphic to G k+2 and ith Lemma 3.30 e obtain the chordality of G k+2. 27