Applied Mathematics E-Notes, (0), - c ISSN 0-0 Available free at mirror sites of http://wwwmaththuedutw/ame/ Combiatio Labeligs Of Graphs Pak Chig Li y Received February 0 Abstract Suppose G = (V; E) is a simple, coected, udirected graph with vertices ad m edges Let f be a bijectio from V oto f; ; :::; g which labels the vertices of G The vertex-labelig f iduces a edge-labelig f C of G as follows: a edge uv E with f(u) > f(v) is assiged the label f C (uv) = f(v) f(u) If the edge labels of G are pairwise distict, the we say G is a combiatio graph I this paper, we will show that complete k-ary trees, wheel graphs, Peterse graphs GP (; ); GP (; ), grid graphs ad certai caterpillar graphs are combiatio graphs We will also show that, except for several special cases, complete bipartite graphs are ot combiatio graphs Itroductio Suppose G = (V; E) is a simple, coected, udirected graph with vertices ad m edges Let f be a bijectio from V oto f; ; :::; g which labels the vertices of G The vertex-labelig f iduces a edge-labelig f C of G as follows: a edge uv E with f(u) > f(v) is assiged the label f C (uv) = f(u) f(v) The labelig f C is called the combiatio labelig of G iduced by the labelig f Whe the combiatio labelig f C is ijective, we say that it is a valid combiatio labelig If the graph G has a valid combiatio labelig, the we say G is a combiatio graph The study of graph labeligs has bee ad cotiues to be a popular topic of graph theory The dyamic survey by Gillia [] shows the diversity of graph labeligs Graceful labeligs are similar to combiatio labeligs A graceful labelig of a simple graph G = (V; E) is a labelig of its vertices with distict itegers from the set f0; ; :::; jejg, such that each edge is uiquely ideti ed by the absolute di erece betwee its edpoits Graceful labeligs have bee extesively studied A well-kow cojecture of graceful labeligs, kow as the graceful tree cojecture, states that all trees have graceful labeligs For a recet survey, see [] By comparig the rage of allowable (iduced) labels o the edges of a graceful labelig versus that of a combiatio labelig, if a graph has both a valid combiatio labelig ad a graceful labelig, it would see that it would be more di cult to d a graceful labelig Fially, it is kow that K ; has a graceful labelig but o valid combiatio labelig, while the -cycle has a valid combiatio labelig but o graceful labelig N Mathematics Subject Classi catios: 0C y Departmet of Computer Sciece, Uiversity of Maitoba, Wiipeg, Maitoba, Caada RT
P C Li I this paper, we cocetrate o the labelig of vertices of a graph which iduces a combiatio labelig o the edges of the graph This problem was itroduced by Hedge ad Shetty [] i 00 I this paper, we will study combiatio labeligs for several classes of graphs ad aswer some questios that were posed by i [] More speci cally, we will show that complete k-ary trees, wheel graphs, geeralized Peterse graphs GP (; ); GP (; ), ad grid graphs are combiatio graphs I additio, we will show that, except for some special cases, complete bi-partite graphs are ot combiatio graphs I Sectio we will show that full k-ary trees, wheel graphs, geeralized Peterse graphs GP (; ); GP (; ), ad grid graphs are combiatio graphs I Sectio, we show that complete bi-partite graphs are ot combiatio graphs, except for a few special cases Classes of Combiatio Graphs I this sectio, we will study several classes of graphs ad show that they are combiatio graphs We begi with rooted trees Trees A rooted tree is a tree where oe of the vertices (or odes) is distiguished from the other This distiguished vertex is kow as the root of the tree The odes of a tree ca be categorized as either o-leaf odes or leaf odes A ode is a leaf ode if it has degree Otherwise, it is a o-leaf ode The depth of a vertex i a rooted tree is the umber of edges o the path from the root to the vertex The height of a tree is the largest depth of ay leaf ode A k-ary tree is a rooted tree where each ode has at most k childre A complete k-ary tree is a k-ary tree where each o-leaf ode has exactly k childre ad the leaf odes have the same depth Our approach will be to show that a rooted tree with the property that all leaf odes have the same depth is a combiatio graph This immediately implies that a complete k-ary tree is a combiatio graph We begi with a simple, useful fact that ca easily be proved by algebraic maipulatios > LEMMA If > k > 0, the k+ k LEMMA Let T be a rooted tree with the property that the depth of ay two leaf odes are the same The T is a combiatio graph PROOF Let T be a rooted tree satisfyig the assumptios stated i the lemma We may assume that T has at least three vertices sice a tree cosistig oe or two odes is a combiatio graph We will d a assigmet f of labels for the odes of T so that the iduced edge labels of T are pairwise distict To label the odes, we will visit ad label (usig the positive itegers) the odes usig a breadth- rst traversal startig at the root such that: the smallest available value is used to label the curret ode beig visited, ad
0 Combiatio Labeligs of Graphs if the depth of two o-leaf odes u ad v are the same ad f(u) < f(v), the the labels assiged to the childre of u are less tha the labels assiged to the childre of v Note that this labelig process does ot ecessarily eed to a uique labelig sice sibligs ca be labeled i ay order As the labels are assiged i a breadth- rst maer, the label of a ode at depth k is smaller tha the label of a ode at depth k + Figure illustrates oe labelig costructed by the labelig process o a give tree Cosider a edge e = uv i the tree T, where f(u) < f(v) are the labels assiged f(v) to the two edpoits of e The label iduced o this edge is f(u) Note that u is the paret of v Suppose the o-leaf ode u has childre v ; v ; :::; v k 0 with f(v ) < f(v ) < < f(v k 0 ) ad k 0 Note that by labelig process, it must be that f(v ) + = f(v ); f(v ) = f(v ) + ; :::; f(v k 0 ) + = f(v k 0 ) By Lemma, f(v 0 k ) f(u) > f(v 0 ) k f(u) > > f(v ) f(u) Therefore, edges betwee a paret ad its sibligs have distict labeligs Now cosider two odes u; w havig the same depth ad f(w) = f(u) + Suppose ode u is a o-leaf ode As all leaf odes have the same depth, the ode w must also be a o-leaf ode By the labelig process, the childre of u must be labeled a; a + ; :::; a + l ad the childre of w must be labeled a + l + ; a + l + ; :::; a + l + m a+l for some a ad l; m By Lemma, f(u) < a+l+ f(u)+ = a+l+ f(w) Therefore all the edges of the same level of the tree are pairwise distict Fially, cosider two o-leaf odes u; w where the depth of u is d, the depth of w is d +, for some d, such that f(u) is the largest label assiged to odes of depth d ad f(w) is the smallest label assiged to odes of depth d + The we see that f(w) = f(u) + Suppose the childre of u are labeled a; a + ; :::; a + l for some a ad l The the childre of w are a + l + ; :::; a + l + m for some m By Lemma a+l, f(u) < a+l+ f(w) This shows that the edges at level d have labels less tha those at level d + Combiig these three results, we see that the tree T is a combiatio graph Lemma immediately implies that complete k-ary trees are combiatio graphs We state this i the followig Corollary THEOREM The complete k-ary tree is a combiatio graph Caterpillars We ow cosider aother class of trees called caterpillars A tree is a caterpillar if, upo removig all leaves ad their icidet edges, a path is left This path is called the cetral path of the caterpillar graph Note that i the caterpillar, the cetral path ca be exteded to a loger path sice each edpoit of the path must be adjacet to a vertex i the caterpillar Let us call this path the exteded cetral path of the caterpillar We will call a edge that is ot o the exteded cetral path of a caterpillar a leg We begi by showig that if a caterpillar has eough legs, the it is a combiatio graph To do this we start with a simple lemma that ca be veri ed through algebraic
P C Li 0 Figure : A labelig of a ode rooted tree maipulatio LEMMA If l; m 0 ad + (l + m), the l < l+m THEOREM Let G = (V; E) be a caterpillar with exteded cetral path P cosistig of p vertices If G has at least p vertices, the T is a combiatio graph PROOF We partitio the vertex set of V ito two smaller sets A ad B by rst dividig the path P ito two (disjoit) sub-paths Q ad R of equal or almost equal legth The we place a vertex v ito A if v Q or v is adjacet to a vertex o Q Otherwise, we place v ito B At least oe of G[A] or G[B] cotai at least (p )= = p edges that are ot edges of P, where G[A](G[B]) deote the subgraph of G iduced by the vertex set A(B) Without loss of geerality, assume G[A] has this property We ow costruct a labelig f of the vertices of V Label the path P startig from oe ed to the other ed with labels ; ; :::; p so that the vertex that is labeled with the value is the ed-vertex of P which belogs to A Now label the remaiig vertices of G that are ot o the path P so that if u; v are ot o P, up ; vp E, p ; p P ad f(p ) < f(p ), the f(u) < f(v) This ca be accomplished by startig at the ed of P label with value, ad movig alog the path P As a leg is ecoutered, we label the vertex of the leg that is ot o the path with the ext available value We claim that this labelig is a combiatio labelig of G We see that the edges o the path P have labels ; ; :::; p ad the smallest edge p+ label of a edge ot o the path is at least > p It is clear that the edge labels of ay two edges i G[A] but ot o P satis es Lemma ad therefore are pairwise distict Fially, the smallest label assiged to a leaf ode i B but ot o P is at least p + (p ) + = p = (p ) I G[B], the label p is the largest label assiged to a vertex o P that ca be adjacet to vertices ot o P Therefore, Lemma ca
Combiatio Labeligs of Graphs be applied to the labels of the legs of the etire graph G to show that the legs of G have pairwise distict labels THEOREM Let G = (V; E) be a caterpillar with exteded cetral path P cosistig of p vertices If each vertex of P, except for its two edpoits, is adjacet to at least oe vertex that is ot o P, the G is a combiatio graph PROOF Start at oe ed of the path P ad label the edpoit Follow the path ad label each vertex visited with the ext available label To label the vertices that are ot o P, use the labelig scheme as i Theorem If f(u) is the label of a vertex u ot o P that is adjacet to a vertex v o P with label f(v), the f(u) f(v) To see this, ote that the smallest value that f(u) ca be is p + f(v) Sice p v, we have f(u) p + f(v) f(v) Applyig Lemma, we see that G is a combiatio graph Geeralized Peterse Graph GP (; k) Suppose k; are positive itegers such that > k The geeralized Peterse graph, deote by GP (; k), is the simple graph with vertices u ; u ; ::; u ; v ; v ; :::; v ad edges u i u i+ ; v i v i+k ; u i v i ; i, where the idexes are take modulo We will show that GP (; ) ad GP (; ) are combiatio graphs LEMMA If the + THEOREM If, the GP (; ) is a combiatio graph PROOF Figure gives a valid combiatio labelig for GP (; ) Therefore, assume that Label the vertices of GP (; ) as follows: f(u ) = ; f(u ) = ; :::; f(u ) = ; f(u ) = ; f(u ) = ; f(v ) = + ; f(v ) = + ; :::; f(v ) = ; f(v ) = ; f(v ) = We claim that this is a valid combiatio labelig of GP (; ) The edges u i u i+ ; i have labels ; ; :::; ; The edges vi v i+ ; i have labels +; :::; ; ; ; = The edges ui v i+ ; i have labels ; + ; :::; By Lemma, + By Lemma, +i i < +i+ i+ I additio, it is easy to see that if, the + Therefore, the edge labels ; ; :::; ; + ; ; + ; + ; :::; ; ; ; are all distict ad are listed i icreasig order LEMMA If, the If =, the = LEMMA If, the + < < + If, the < + THEOREM For, the GP (; ) is a combiatio graph PROOF Figure gives valid combiatio labeligs for GP (; ), where Therefore, assume that Label the vertices of GP (; ) as follows: f(u ) = ; f(u ) = ; :::; f(u ) = ; f(u ) = ; f(u ) = ; f(v ) = ; f(v ) = + ; :::; f(v ) = ; f(v ) = ; f(v ) = We claim that this is a valid combiatio labelig of GP (; ) The edges u i u i+ ; i have labels ; ; :::; ; + The edges v i v i+ ; i have labels ; + ; :::; ; + ; ; ad The edges ui v i+ ; i have labels ; + ; :::; Note that
P C Li Figure : A valid combiatio labelig of GP (; ): + = ; = ; = ad = We claim that we ca order the edge labels i mootoe icreasig order We ca order this smallest edge labels as < < < + < < where the last iequality holds as > Cotiuig, we have + + By Lemma, + we have either < + < < where the last iequality follows from Lemma, or + + < + < < Fially, we have < < This gives a sequece of strict iequalities ivolvig each edge label Therefore, GP (; ) is a combiatio graph Wheel Graphs Let be a positive iteger greater tha A wheel graph o + vertices is a graph cosistig of a cycle of legth ad a vertex ot o the cycle that is adjacet to every vertex o the cycle We deote this graph by W I [], it was cojectured that for all, W is a combiatio graph We will show that this cojecture is true We begi with some simple, useful results that ca be veri ed by algebraic maipulatios LEMMA If is a eve umber, the = = LEMMA If 0 is a eve umber, the =+ =+ I additio, if 0 is a eve umber, the =+ < =+ LEMMA If is a odd umber, the b=c < b=c We ow proceed to label the wheel graph W We will give a labelig that almost works ad the modify it slightly to so that it gives a valid combiatio labelig of W THEOREM If, the W is a combiatio graph PROOF Valid combiatio labeligs for = ; were give i [] Let us assume that Deote the cycle of legth of W by C Let x be the vertex that is ot
Combiatio Labeligs of Graphs 0 0 0 0 Figure : Valid combiatio labeligs for GP (; ) for : i C ad is adjacet to each vertex of C Deote the vertices of C by v 0 ; v ; :::; v where v i is adjacet to v i+ modulo We ow give a labelig of the vertices of W Label vertex x with value O the cycle C, label v 0 with, v with, v with, etc I geeral, after labelig vertex v i with value k, we skip over vertex v i+, ad label v i+ with value k + if it has ot already bee labeled If v i+ has already bee labeled, the we label v i+ with value k + The idexes are take modulo Let us deote this labelig by f f(v Uder this labelig there exists (at least oe) i such that i) f(v i+) = f(v i) f(v i ) Whe f(v ) is odd, we have f(v ) = Therefore f(v ) = d=e = b=c = f(v ) f(v ) However, this is the oly occurrece because for ay vertex whose label l is greater tha b=c +, its eighbors o C have labels l b=c ad l d=e The oly l value of l which satis es l b=c = l l d=e is l = Usig a similar argumet for f(v whe is eve, we see that i) f(v i+) = f(v i) f(v i ) happes oly whe i = ad f(v ) = We make a slight modi catio to the labelig f by performig the followig swaps: If is odd, we swap the labels ad i the labelig f If is eve, we swap the labels ad i the labelig f Let us deote this ew labelig by g We claim that g is a combiatio labelig l of W It is clear that two adjacet edges o C do ot have labels l k ad l k for l > k because of the modi catios made above Figure gives examples for = 0; I the case where is odd, the edge labels of W, iduced by g are ; ; :::; ad b=c+ ; b=c+ ; :::; b=c ; b=c ; b=c ; b=c+ ; b=c ; b=c ; b=c+ ;
P C Li 0 0 Figure : Combiatio labeligs for W ad W 0 : b=c+ ad b=c+ Note that b=c+ = b=c If remove b=c ; b=c from the sequece above, the remaiig values are all distict ad i fact, < < + < b=c+ b=c+ b=c+ < < b=c b=c < b=c+ < b=c < b=c+ (= b=c ) < b=c+ By Lemma, b=c+ < < < Sice < < b=c b=c b=c b=c b=c b=c+, all the edge labels are distict, whe is odd We ow cosider the case whe is eve The edge labels of W,iduced by g are =+ ; ; :::; ; ; =+ ; =+ ; =+ ; :::; = ; = ; = ; = ; =+ ; =+ ad If we remove = ; = ad from this list, the remaiig values are clearly distict ad < + < =+ =+ < =+ < < = = = = < = = =+ < = <= = = =+ Note that as = = < = ad by Lemma, = < =, all edge labels except for possibility are distict By Lemma =+, =+ for all eve 0 For = 0; ; ; ;, we have < < =+ Therefore all the edge labels are distict i W =+ Grid Graphs = ; Let k; be positive itegers with k Let G = (V; E) be the k grid graph More precisely, V = f(i; j) : 0 i k ; 0 j g ad E = ff(i; j ); (i; j )g : 0 j ; j = j + g [ ff(i ; j); (i ; j)g : 0 i k ; i = i + g Aother way of costructig the k grid graph is to take the Cartesia product of the paths P k ad P We claim that for large eough values of k ad, the k grid graph is a combiatio graph To prove this, we begi with a useful fact LEMMA 0 Let k If d k +p k k+ e, the k < + THEOREM Let k If d k +p k k+ e, the the k grid graph is
Combiatio Labeligs of Graphs a combiatio graph PROOF Label the vertex (i; j) with the value i + j + Thethe edges have + iduced labels i the set f; ; :::; k ; k; ; + ; :::; k ; k gf; + + ; :::; (k ) + g Clearly, the labels ; + ; :::; k for a icreasig sequece ad therefore are pairwise distict Sice d k +p k k+ e, Lemma 0 implies = + > k Therefore, the edge labels are pairwise distict + Other Results We state several related results LEMMA Let G be a graph with vertices If G is a combiatio graph, the at most oe vertex of G has degree PROOF Suppose there are at least two vertices that have degree i G ad G is a combiatio graph Cosider a valid combiatio labelig Let x < y be vertex labels of two vertices of degree Suppose y The y is adjacet vertices labeled y ad y But y = y, which cotradicts assumptio that G is a combiatio graph Therefore y implyig x = ; y = The both x; y are adjacet to the vertex labeled As =, this cotradicts assumptio that G is a combiatio graph Therefore, at most oe vertex of G ca have degree This immediately implies that K is ot a combiatio graph wheever The proof of Lemma also shows that if a combiatio graph has a vertex of degree, the label of that vertex must be or We ow show that some combiatio graph o vertices with the maximum umber of edges possible must cotai a vertex of degree whose label is LEMMA Let m be the maximum umber of edges i ay combiatio graph with vertices The there is a combiatio graph G with vertices ad m edges such the vertex labeled with value is adjacet to all the other vertices PROOF Suppose that G is a combiatio graph with vertices, m edges ad the vertex v labeled with value does ot have degree The, let the vertices that are ot adjacet to v have labels a ; a ; :::; a k, where k Remove from G all edges whose iduced edge labelig belogs i fa ; a ; :::; a k g There are at most k such edges, as G is a combiatio graph Fially, add edges to G so that v has degree The resultig graph is still a combiatio graph Sice the origial graph was a combiatio graph ad has maximum umber of edges possible, the umber of edges removed must be k We ca use Lemma to show that ay combiatio graph with vertices ca have at most edges I the cotrary, suppose G is a combiatio graph with vertices ad edges By Lemma there must exists a graph H with vertices ad edges such that the vertex labeled with value is adjacet the other vertices The oly remaiig edges that are permissible the edges ; ; ; ; But sice = ad =, at most of these edges ca be i the graph H give a total of edges, which is a cotradictio A similar argumet ca be applied to obtai the followig boud from []
P C Li m ( = if is eve ( )= if is odd: () I [], it was show that K r;r is ot a combiatio graph for r We ow geeralize this for complete bipartite graphs K l;k Note that if a graph G = (V; E 0 ) is ot a combiatio graph, the it is clear that if we add additioal edges to E 0 it will ot be a combiatio graph We record this i the followig lemma LEMMA Suppose G = (V; E) ad E 0 E If (V; E 0 ) is ot a combiatio graph, the G is ot a combiatio graph THEOREM Let K l;k = (A; B) be the complete bipartite graph with k elemets i the partite set A, ad l elemets i the other partite set B The K l;k is a combiatio graph if ad oly if k = or l = or k = l = PROOF The case where k = l = the cycle of legth which is clearly a combiatio graph Suppose k = (or if l = ) ad let A deote the partite set with oe vertex Label the loe vertex of the partite set A with value ad label the vertices i the other partite set with values ; ; :::; l + Clearly this is a valid labelig Suppose l ad k > l We will show that K l;k is ot a combiatio graph To do this, suppose to the cotrary that K l;k is a combiatio graph where the vertices of the graph is labeled usig a valid combiatio labelig The the vertices with label ad l + k must be i the same partite set For if ot, the without loss of geerality suppose A, l + k B This forces vertex with label l + k to be i B, which i tur forces vertex with label l + k to be i B, ad so o Therefore, the partite set A cotais oly oe vertex, the vertex with label This cotradicts our assumptio that jaj We ow have two scearios: ; l + k A or ; l + k B Case : Suppose ; l + k A If the vertex with label l + k B, the usig a argumet similar to the oe above, we have ; ; :::; l + k B This implies that k+ l = As k >, = k+ k ad ; k B, we have two edges with the same label, which is a cotradictio Therefore, it must be that l + k A By repeatedly applyig this argumet ad the assumptio that jbj = k, we see that the labels i A are f; k + ; k + ; :::; l + kg ad the labels i B are f; ; :::; k + g As k > ad k+ = k+, there are two edge labels with the same value, which is a cotradictio k Therefore, case leads to a cotradictio Case : Suppose ; l + k B The, usig a argumet similar to that of case, we have that A cotais vertices with labels ; ; :::; l + ad B cotais vertices with labels ; l + ; :::; l + k The labels l; l + A Sice k > l, l + l + k implyig l+ that l + B This alog with the fact that l = l+ l+ implies that two edges have the same labelig, which is a cotradictio Therefore, case also leads to a cotradictio COROLLARY Suppose G is a complete k-partite with partite sets A ; A ; ::; A k where k ad ja i j for i = to k The G is ot a combiatio graph
Combiatio Labeligs of Graphs PROOF Costruct a graph H from G by removig all edges betwee A i ; A j where < i = j k Partitio the vertices of H ito two sets A ad [ k i= A i The graph H is a complete bipartite graph By Theorem, it is ot a combiatio graph By Lemma, G is ot a combiatio graph We would like to ed by statig the followig ope problem: Are all trees combiatio graphs? Based o istaces that we have cosidered, which all tured out to be combiatio graphs, we believe ad cojecture that all trees are combiatio graphs Ackowledgmet Research supported by NSERC Discovery Grat 0-0 Refereces [] M Edwards ad L Howard, A survey of graceful trees, Atlatic Electroic Joural of Mathematics, (00), 0 [] J A Gallia, Graph labelig, Electroic Joural of Combiatorics, DS, pages [] S M Hedge ad S Shetty, Combiatorial labeligs of graphs, AMEN, (00),