Graceful Labelings of Pendant Graphs

Transcription

1 Rose-Hulma Udergraduate Mathematics Joural Volume Issue Article 0 Graceful Labeligs of Pedat Graphs Alessadra Graf Norther Arizoa Uiversity, ag@au.edu Follow this ad additioal works at: Recommeded Citatio Graf, Alessadra (0) "Graceful Labeligs of Pedat Graphs," Rose-Hulma Udergraduate Mathematics Joural: Vol. : Iss., Article 0. Available at:

2 Rose- Hulma Udergraduate Mathematics Joural Graceful Labeligs of Pedat Graphs Alessadra Graf a Volume, No., Sprig 0 Sposored by Rose-Hulma Istitute of Techology Departmet of Mathematics Terre Haute, IN 03 mathjoural@rose-hulma.edu a Norther Arizoa Uiversity

3 Rose-Hulma Udergraduate Mathematics Joural Volume, No., Sprig 0 Graceful Labeligs of Pedat Graphs Alessadra Graf Abstract. I, Alexader Rosa itroduced a ew type of graph labelig called a graceful labelig. This paper will provide some backgroud o graceful labeligs ad their relatio to certai types of graphs called pedat graphs. We will also preset ew results cocerig a specific type of graceful labelig of pedat graphs as well as further areas of research. Ackowledgemets: I would like to give a special thaks to my metor Jeff Rushall for all of his help ad guidace, ad to Ia Douglas for sharig his programmig expertise.

4 Page RHIT Udergrad. Math. J., Vol., No.. Itroductio From elemetary school studets to uiversity seiors, most studets have some exposure to graphs. The phrases graph the fuctio ad based o the graph are a commo occurrece i most stadardized math courses. They cojure up images of coordiate axes ad itersectig curves, which ofte idicates that calculatios ivolvig slopes ad area are o the horizo. As the oly exposure studets have to graphs occur i geometry ad calculus-related courses, which oly study graphs of this form, most studets believe that all graphs ivolve axes ad fuctios. However, that is simply ot true. I fact, most types of graphs do ot have coordiate axes or ivolve a specific fuctio. Outlies for essays, the game board for The Settlers of Cata R, ad fire evacuatio route plas are all examples of graphs without these characteristics. To uderstad why, cosider the defiitio of a graph. Defiitio. A graph is a collectio of poits together with a collectio of lies coectig some subset of those poits. The poits are called vertices ad the lies are called edges. Notice that this defiitio does ot icite a image studets typically associate with graphs. Istead, it seems to be more aki to childhood games ad activities, such as coectthe-dots puzzles ad the Chiese board game Go. A closer look at the iformal defiitio of a graph of a fuctio, however, shows otherwise. Defiitio. The graph of a fuctio f is the set of ordered pairs showig the values take by f over the domai of f. Thus, a graph of a fuctio ca be thought of as a graph of vertices with o edges coectig them! The curve does ot look like a set of distict vertices because there are ifiitely may vertices very close together. Therefore, to a graph theorist, all of the images i Figure are cosidered to be graphs. Figure I the ext few sectios, we will discuss some of the basic properties of graphs, icludig graph labeligs, ad how these ca be assiged to differet graphs. We will the defie a graceful labelig ad study a few examples to uderstad whe such a labelig exists ad how we ca produce these labeligs for certai types of graphs. The remaider of the paper will the focus o pedat graphs ad their graceful labeligs. To coclude this paper, we will see some ew results as well as provide some ope questios.

5 RHIT Udergrad. Math. J., Vol., No. Page. Graph Theory Basics The study of the properties ad structures of graphs is called graph theory. Ideas ad results from graph theory have bee used to help model a variety of systems, from egieerig blueprits to outliig a college essay. Differet properties may be more valuable for modelig certai systems tha others, so it is importat to categorize graphs i terms of their structure. Example. Cosider the graphs i Figure. A B C Figure Although each graph cotais vertices ad edges (ote that each arrow directio of graph B determies a edge, so the segmet coectig the top ad bottom vertices is really edges), some differeces betwee the graphs are apparet. For example, graph C has a vertex that is ot coected by a edge to ay other vertex i the graph. We say graph C is ot coected. To differetiate these graphs further, ote the followig defiitios. Defiitio. A graph is coected if there exists a sequece of edges (called a path) that ca be trasversed from ay vertex to ay other vertex i the graph. Defiitio. A graph is directed if the edges of the graph are assiged a iitial ad termial vertex. This is usually idicated by a arrowhead located somewhere alog the edge that poits to the termial vertex. Defiitio.3 A graph is simple if it is a udirected graph that cotais either loops or multiple edges coectig the same vertices. Defiitio. A pseudograph is a graph that may cotai loops (i.e. a edge coectig a vertex to itself) or multiple edges coectig the same pair of vertices. Defiitio. Two vertices are adjacet if there is a edge coectig them. Example. (cot.) To classify the graphs i Figure, cosider whether each graph has multiple edges coectig a pair of vertices or a loop, if the edges are directed, ad whether the graph is coected. Sice graph C has a loop, three edges coectig the same vertices, ad o directios idicated alog its edges, we ca classify graph C as a udirected pseudograph that is ot coected. Similarly, the edges of graph B are directed, yet there is still a path from ay vertex of B to every other vertex of B. As there are two edges coectig the top ad bottom vertices, graph B is a coected, directed pseudograph. Fially, graph A cotais o loops or multiple edges ad each vertex of A is adjacet to every other vertex

6 Page 0 RHIT Udergrad. Math. J., Vol., No. of A. Therefore graph A is a simple, coected graph. Based o these properties ad others, graph theorists orgaize graphs ito families of graphs that share some predetermied traits. These families ofte have special ames, such as tree graphs, wheel graphs, ad sprayig pipe graphs. Example. defies ad explais aother family of graphs, called complete graphs, which we will use later. Example. Cosider the graphs i Figure 3. A B C Figure 3 Graphs A, B, ad C are all members of the complete graph family. To be a member of this family, every pair of vertices i the graph must be coected by a edge. These graphs are referred to as K where is the umber of vertices i the graph. Sice graph A has oly oe vertex, there is o pair of vertices that ca be coected. Hece the loe vertex with o edges is the complete graph K. Graphs B ad C are the complete graphs K 3 ad K respectively. However, it should be oted that a graph ca belog to more tha oe family. For example, graph B is also the cycle o 3 vertices. So graph B is both the complete graph K 3 ad the cycle graph C Graph Labeligs Alog with its iheret structure, a graph ca be assiged additioal properties. This is ofte accomplished by assigig labels to the vertices ad edges of the graph. Example 3. Suppose Figure is modelig the populatio of frogs i a certai pod. Figure I its curret state, the graph provides very little iformatio about the size of the frog populatio ad how it chages over time. If we were to label the directed edges with the birth rate ad death rate of the populatio ad the cetral vertex with the curret frog populatio, Figure would become a more useful model of the frog populatio. Defiitio 3. A graph labelig is a assigmet of itegers to the vertices, edges, or both of a graph that is subject to certai coditios. The suggested chages to Figure are a example of a graph labelig. Specifically,

7 RHIT Udergrad. Math. J., Vol., No. Page the edges ad vertices would be assiged itegers that are subject to the coditios the directed edges are labeled with the birth rate ad death rate of the populatio ad the cetral vertex is labeled with the curret frog populatio, respectively. Example Figure Figure depicts aother graph labelig where edges are labeled with the product of the labels of the vertices they coect. Note that two edges are assiged the same label. This is acceptable so log as the coditios chose for the labelig do ot require labels to be uique. Example 3.3 Cosider the graph i Figure. The vertices ad edges have bee labeled accordig to the followig rules:. The edges must be labeled usig each of the itegers through oce.. The vertices must be labeled usig each of the itegers 0 through o more tha oce. 3. Every edge must be the positive differece of the vertex labels it coects Figure The graph labelig show i Example 3.3 is called a graceful labelig.

8 Page RHIT Udergrad. Math. J., Vol., No.. Graceful Labeligs I, Alexader Rosa [] itroduced a ew type of graph labelig which he amed a β-labelig. As Rosa reflects i [], it was believed these labeligs would help solve Rigel s cojecture, which ivolves decomposig a complete graph ito isomorphic subgraphs. This type of labelig has sice bee reamed a graceful labelig ad has bee used to model situatios i may fields, icludig radio astroomy, circuit desig, codig theory, ad X-ray crystallography. It is formally stated i Defiitio.. Defiitio. A simple, coected graph with q edges is said to be graceful if its vertices ad edges are labeled such that: (a) each vertex is assiged a distict iteger from the set {0,..., q}, (b) each edge is assiged a distict iteger from the set {,..., q} such that the label is equal to the absolute value of the differece of the two vertices the edge coects. By returig to the labelig rules used i Example 3.3, it ca be see that the vertices are assiged distict itegers from the set {0,...,} (by rule ) ad the edges are assiged distict itegers from the set {,...,} (by rule ) so that each edge is the positive differece of the vertex labels it coects (by rule 3). The graceful labelig show i Figure is by o meas the oly graceful labelig for the graph show. I fact, this particular graph has distict graceful labeligs. Other graphs have more tha 0,000 distict graceful labeligs. Yet may graphs caot be gracefully labeled. Example. Recosider the graphs from Figure (show below). A B C Figure The defiitio of a graceful labelig requires a simple, coected graph. Thus, graphs B ad C do ot have graceful labeligs sice either is a simple, coected graph. Graph A, however, might have a graceful labelig. I fact, a example of such a labelig is show i Figure.

9 RHIT Udergrad. Math. J., Vol., No. Page Figure Fidig a graceful labelig for a graph is usually accomplished through trial ad error. Ivestigators assig the vertices of the graph eligible labels ad costatly check the differeces betwee the labels of adjacet vertices to esure that the iduced edge labels are distict. This process is tedious ad does ot efficietly fid every possible graceful labelig of a graph. Also, graph theorists ted to study families of graphs rather tha idividual graphs. These families are ofte ifiite sice oe ca ofte add vertices ad edges to a member of a family to create a larger graph that still shares the same defiig characteristics. Thus, sice there are usually ifiitely may members of a family of graphs, provig that every graph of a certai family has a graceful labelig caot be accomplished by fidig examples of such a labelig for oly fiitely may family members. Istead, researchers develop fuctios that, whe give a iput of q, retur a ordered set of itegers. These itegers, whe used as vertex labels for the q vertices of a graph i the specified order, produce a graceful labelig for the graph. By provig that the fuctio s output will always iduce a graceful labelig for a graph with specified characteristics (such as belogig to a particular family of graphs), it the follows that all graphs that satisfy those characteristics have a graceful labelig. Example. Hebbare s formula [] for graceful labelig cycles C where 0, 3 (mod ) is the sequece: 0,,,,,,..., where the iteger + is omitted. Figure shows the graceful labelig of C that follows this formula Figure

10 Page RHIT Udergrad. Math. J., Vol., No. Formulas like the oe show i Example. ofte rely o properties specific to the family of graphs as well as the properties of graceful labeligs. For istace, a -cycle (or C ) has exactly vertices ad edges. Sice the vertices are labeled with itegers from the set {0,..., }, which has + elemets, exactly oe of the possible vertex labels will ot appear i the graceful labelig. Also, a graceful labelig of the -cycle requires the edges to be labeled with distict itegers from the set {,..., }. This forces the vertex labels 0 ad to be used i the labelig ad to be assiged to adjacet vertices (sice 0 is the oly eligible differece that iduces a edge label of ). These ad other properties help ivestigators develop ew formulas for families of graphs that share these traits.. Pedat Graphs ad Their Graceful Labeligs Defiitio. The coroa G G of graphs G ad G is the graph obtaied by takig oe copy of G, which has p vertices, ad p copies of G, ad the joiig the ith vertex of G by a edge to every vertex i the ith copy of G. Example.: The followig are two examples of the coroas of two graphs. C K C K Figure Defiitio. A pedat graph is a coroa of the form C K where 3. Example. Figure ad the secod graph i Figure are examples of pedat graphs. Sice the cycle C has vertices ad the complete graph K has vertex, a pedat graph cosists of each vertex of the cycle beig joied by a edge to a sigle vertex i the correspodig copy of K. The vertices of the copies of K are the referred to as pedat vertices. Thus, a pedat graph has vertices ( from the cycle ad from the copies of K ) ad edges ( from the cycle ad edges joiig the cycle to the copies of K ). As evideced by the graceful labelig show i Figure, some pedat graphs have graceful labeligs. Graceful labeligs of pedat graphs were ivestigated by Roberto Frucht [] i. He foud that these labeligs fell ito oe of three categories.

11 RHIT Udergrad. Math. J., Vol., No. Page Defiitio.3 A graceful labelig of a pedat graph is of the:. first kid if the labels 0 ad are assiged to adjacet vertices of the -go.. secod kid if is assiged to a pedat vertex. 3. third kid if 0 is assiged to a pedat vertex. Note that 0 ad caot both be assiged to pedat vertices. This is because, like cycle graphs, i order to get a edge differece of, the vertex labels 0 ad must be assiged to adjacet vertices (sice 0 is the oly eligible differece that iduces a edge label of ). Sice o two pedat vertices are adjacet, 0 ad caot both be assiged to pedat vertices. Example.3: Figure 0 provides examples of graceful labeligs of the first, secod, ad third kid First Kid Secod Kid Third Kid Figure 0 Frucht was able to prove that all pedat graphs have a graceful labelig usig separate formulas for differet cases (oe for each cogruece class of (mod )). However, his formulas oly produced graceful labeligs of the secod kid. Frucht was able to fid eough examples to cojecture that graceful labeligs of the first kid exist for all pedat graphs

12 Page RHIT Udergrad. Math. J., Vol., No. as well, but he was uable to prove this claim. I a 03 paper [3], fuctios producig graceful labeligs of the first kid for pedat graphs with 3, (mod ) were proved to exist (see Theorem. ad Example.). Theorem. Let {v, v,..., v } be the set of cycle vertices ad {u, u,..., u } be the set of pedat vertices for the pedat graph C K where v i is adjacet to u i, v i, ad v i+ if i {, 3,..., }. Note that v is adjacet to u, v, as well as v ad that v is adjacet to u, v, ad v. If (mod ), >, the followig fuctio assigs a vertex label f(v i ) to the cycle vertex v i ad assigs vertex label f(u i ) to the pedat vertex u i. The resultig labelig is a graceful labelig of the first kid for C K. 0 if i = i + if i =, 3,,..., f(v i ) = i + if i =, +, +,..., i if i =,,,..., f(u i ) = if i = if i = i + if i = 3 +, 3 + 3, 3 +,..., i + if i = 3, 3 +, 3 +,..., 3 i + if i = 3 i + if i =, +, + 3,..., 3 i if i =,,,..., 3 3 i + if i =, 3,,..., 3 Example.: Figure shows the graceful labelig of the first kid for a pedat graph with = 0 produced by Theorem.. Note that ay cycle vertex ca be chose as v ad the labelig ca proceed i either a clockwise or couterclockwise fashio without chagig the labelig. I this example, v (which receives the label 0) occurs at the bottom of the cycle ad the labelig proceeds i a couterclockwise fashio.

13 RHIT Udergrad. Math. J., Vol., No. Page New Results 3 3 Figure I my ivestigatio, I have cotiued the efforts of provig Frucht s cojecture that every pedat graph has a graceful labelig of the first kid. Although difficult, the process became easier with the help of a Mathematica program I developed with the help of Ia Douglas. Whe give a iteger, this program was able to fid ad list all graceful labeligs of a pedat graph of cycle size. The program was the modified to accept a partial graceful labelig i additio to the size so that the graceful labeligs retured followed the provided labelig. This decreased the umber of labeligs the program retured from hudreds of thousads of possible labeligs to i some cases less tha 00 possibilities. The program was used o values of i the same cogruece class (mod ) to idetify patters preset i some of the labeligs for each of these values of. [Note that cogruece classes (mod ) were chose because the graceful labeligs of pedat graphs with cycle sizes ad + appeared to be very similar.] These patters would the be added to the partial graceful labelig used by the Mathematica program ad the program was ru agai for the same values of. This process cotiued util the program retured exactly oe graceful labelig for each tested value. These labeligs were the used to create a fuctio that would produce a graceful labelig for ay value of the same cogruece class (mod ) (or i some cases (mod )). The fuctio was the rigorously tested to esure it was valid for all such before a proof for the fuctio was writte. Theorem. is a example of oe of

14 Page RHIT Udergrad. Math. J., Vol., No. the results my ivestigatio produced. Theorem. Let {v, v,..., v } be the set of cycle vertices ad {u, u,..., u } be the set of pedat vertices for the pedat graph C K where v i is adjacet to u i, v i, ad v i+ if i {, 3,..., }. Note that v is adjacet to u, v, as well as v ad that v is adjacet to u, v, ad v. If (mod ), > 0, the followig fuctio assigs vertex label f(v i ) to the cycle vertex v i ad assigs vertex label f(u i ) to the pedat vertex u i. The resultig labelig is a graceful labelig of the first kid for C K. f(v i ) = f(u i ) = i + if i =, 3,,..., i if i =,,,..., + if i = + 3 i if i = +, +,..., i if i = + 3, +,..., + if i = 0 if i = 3 if i = if i = i if i = 3+ + i if i = i if i =, 3,,..., i + if i =,,,..., 3 i if i = +, ,..., 3 + i if i = +, 3+ +,..., i if i = +, ,..., i if i = +, 3+ +,..., Example.: Figure shows the graceful labelig of the first kid for a pedat graph with = produced by Theorem.. Note that ay cycle vertex ca be chose as v ad the labelig ca proceed i either a clockwise or couterclockwise fashio without chagig the labelig. I this example, v (which receives the label 3) occurs at the bottom of the cycle ad the labelig proceeds i a couterclockwise fashio.

15 RHIT Udergrad. Math. J., Vol., No. Page Figure Proof. First, we shall verify that statemet (a) of Defiitio. is satisfied. As o vertex belogs to more tha oe lie of the fuctio, each vertex is assiged precisely oe label. Through close examiatio of the above fuctio f, it ca be see that each vertex is assiged a label from the set {0,..., }. Specifically, the first lie of f(u i ) assigs the odd itegers from to to uique pedat vertices of the graph. The secod lie of f(v i) assigs the eve itegers from to + to uique cycle vertices of the graph. Thus, all itegers from to + are used i the graph labelig. Similarly, the fourth lie of f(v i ) assigs the odd itegers from + to 3 (sice 3 ( +) = 3, 3 ( +) =,..., 3 ( 3) = +) ad the eighth lie of f(u i ) assigs the eve itegers from 3+ + to (as ( 3+ + ) = 3+, ( 3+ + ) = 3+,..., ( ) = + ). The odd itegers from to + are assiged by the fourth lie of f(u i ) ( 3+( +) =, 3+( +) = +,..., 3 + ( 3+ ) = + ) ad the eve itegers from to are assiged by the the third lie of f(u i ) ( 3 ( + ) =, 3 ( + 3) =,..., 3 ( 3+ ) = 3 + 3). Clearly, the fifth ad sixth lies of f(u i ) assig the odd labels + ad + ad the third lie of f(v i ) ad last lie of f(u i ) assig the eve labels + ad. The seveth lie of f(u i ) assigs the odd labels from + + to 3 (sice + + ( 3+ + ) = + +, + + ( ) = + +,..., + + ( 3) = 3 ) ad the fifth lie of f(v i) assigs the eve itegers from + to 3 ( + + ( + 3) = +, + + ( + ) = +,..., + + ( ) = 3 ). Fially, the ith ad secod lie of f(u i) assig the odd labels from 3 to (sice + =, + = 3,..., ( )+ = 3 +)

16 Page 0 RHIT Udergrad. Math. J., Vol., No. ad the first lie of f(v i ) assigs the eve itegers from 3 + to (as + =, 3 + =,..., ( ) + = 3 + ). Thus, all of the itegers from 0 to are assiged to a uique vertex except the eve label + 3 which is omitted from the labelig. Therefore each vertex must be assiged a uique label, which shows statemet (a) is verified. To verify statemet (b) of Defiitio., we must calculate the differeces of adjacet vertex labels to determie the iduced edge labels. We shall begi by checkig the edge labels iduced by the cycle vertex labels of i ad i +. Whe i < ad i is odd, the edge labels are f(v i) f(v i+ ) = ( i+) (i+) = i, which produces values that start at ad decrease by as i icreases by util i = (which always produces the label ). Whe i < ad i is eve, the edge labels are f(v i+ ) f(v i ) = ( (i + ) + ) (i) = i, which produces values that start at ad decrease by as i icreases by util i = (which always produces the label + ). Therefore the sequetially icreasig path of cycle edges from vertex to vertex have eve edge labels,,..., +. The edge label betwee i = ad i + is f(v i) f(v i+ ) = ( ( ) + ) ( + ) = 3 = 3. Whe < i 3 ad i is odd, the edge labels are f(v i+) f(v i ) = ( ++(i+)) ( 3 (i)) = i+3, which produces values that start at ad icrease by as i icreases by util i = 3 (which always produces the label 3). Whe i is eve ad + i <, the edge labels are f(v i) f(v i+ ) = ( ++(i)) ( 3 (i+)) = i+3, which produces values that start at ad icrease by as i icreases by util i = (which always produces the label ).Therefore the sequetially icreasig path of cycle edges from vertex + to vertex have odd edge labels,,..., 3. The edge label from to is f(v ) f(v ) = ( + + ( )) ( + ) =, from to is f(v ) f(v ) = ( + ) (0) = +, ad from to is f(v ) f(v ) = ( () + ) (0) =. Thus, the cycle edge labels are,,,..., 3,, 3, +, +, +,...,. Next, we shall check the edge labels iduced by the cycle ad pedat vertex labels for each i. Whe i ad i is odd, the edge labels are f(v i) f(u i ) = ( i + ) (i) = i +, which produces values that start at ad decrease by as i icreases by util i = (which always produces the label + ). Whe i is eve ad i, the edge labels are f(u i ) f(v i ) = ( (i) + ) (i) = i +, which produces values that start at 3 ad decrease by as i icreases by util i = (which always produces the label + 3). Therefore the pedat-cycle edges for i have odd edge labels, 3,..., +. The pedat-cycle edge whe i = + is f(v i) f(u i ) = ( + ) ( 3 ( + )) = + ( ) =. Whe + i 3+ ad i is odd, the edge labels are f(u i ) f(v i ) = ( 3 + (i)) ( 3 (i)) = i, which produces values that start at ad icrease by as i icreases by util i = 3+ (which always produces the label 3). Whe i is eve ad + 3 i 3+, the edge labels are f(v i ) f(u i ) = ( + + (i)) ( 3 (i)) = i +, which produces values that start at ad icrease by as i icreases by util i = 3+ (which always produces the label ).Therefore the pedat-cycle edges for eve i where + i 3+ have

17 RHIT Udergrad. Math. J., Vol., No. Page edge labels,,..., ad for odd i where + i 3+ have edge labels,, 0,..., 3. Thus the eve labels from to 3 are iduced. Whe i = 3+, the pedat-cycle edge is assiged f(u i ) f(v i ) = ( + + ( 3+ )) ( + + ( 3+ ) = ad whe i = 3+, the pedat-cycle edge is assiged f(v i ) f(u i ) = ( + + ( 3+ )) ( + ( 3+ )) = 3. The edge labels assiged whe i is odd ad 3+ + i 3 are f(u i ) f(v i ) = ( + + (i)) ( 3 (i)) = i +, which produces values that start at + ad icrease by as i icreases by util i = 3 (which always produces the label ). Similarly, whe i is eve ad 3+ + i are f(v i ) f(u i ) = ( + + (i)) ( (i)) = i, which produces values that start at + 3 ad icrease by as i icreases by util i = (which always produces the label ). Thus the eve labels from + 3 to are iduced. Fially, whe i =, the pedat-cycle edge is labeled f(u i ) f(v i ) = ( 3 ) ( +) = ad whe i =, the pedat-cycle edge is labeled f(u i ) f(v i ) = () (0) =. Thus, the edge labels are uique ad are from the set {,..., }, so statemet (b) is satisfied. Therefore the pedat graph is gracefully labeled. Remark. Graceful labeligs of the first kid exist for those pedat graphs with values of smaller tha the restrictio give i Theorem.. For these small values of, some vertices belog to multiple portios of the piecewise fuctio. A graceful labelig ca be produced usig the fuctio from Theorem. by lettig the first label a vertex is assiged by the formula to be the label of that vertex (i.e. vertex labels are the assiged i the order they are listed i the piecewise fuctio). For example, i the pedat graph = 0, the vertex u correspods to both i = 3+ ad i = +. By followig the rule stated above, this vertex is assiged the label f(u ) = ( 3(0)+ ) = + = 3.. Coclusio ad Ope Questios Whe Alexader Rosa [] itroduced his ew β-labeligs i, he believed they would help solve Rigel s cojecture. Little did he kow that graceful labeligs would be used i a much broader scope. Over the past years, graceful labeligs have bee used to model a variety of situatios i several fields. Fidig graceful labeligs for a specific graph or model, however, is ofte difficult. So, researchers have created formulas that produce a graceful labelig for specific types of models, which saves time ad eergy. Recetly, ew formulas have bee foud to provide graceful labeligs of the first kid for pedat graphs of cycle size, 3, (mod ). Other formulas (which are still beig verified) seem to produce graceful labeligs of the first kid for pedat graphs with cycle sizes, (mod ). No such formulas have yet bee idetified for those of cycle size 0,, (mod ). Other ope questios ivolve adaptig these or other formulas to fid fuctios that produce graceful labeligs for other similar families of graphs. For example, are uicyclic graphs, which are graphs that cotai exactly oe cycle, graceful so log as the cycle C is

18 Page RHIT Udergrad. Math. J., Vol., No. of size 0, 3 (mod )? Could the graceful labelig formulas for pedat graphs (which are a type of uicyclic graph) be adapted to help prove this cojecture? Are there other formulas that produce graceful labeligs for pedat graphs that would be more beeficial i certai applicatios? There are may more uaswered questios cocerig graceful labeligs of graphs. The iterested reader ca fid more cojectures ad results for graceful ad other types of graph labeligs i a survey by Gallia [].. Refereces. R. Frucht, Graceful umberig of wheels ad related graphs, A. New York Acad. Sci., 3() -.. J. Gallia, A dyamic survey of graph labelig, The Electroic Joural of Combiatorics, (00). 3. A. Graf, A ew graceful labelig for pedat graphs, Aequatioes Math., (03).. S. P. R. Hebbare, Graceful cycles, Util. Math, () A. Rosa, O certai valuatios of the vertices of a graph, Theory of Graphs (Iterat. Symposium, Rome, July ), Gorda ad Breach, N. Y. ad Duod Paris () A. Rosa, A discourse o three combiatorial diseases, Desigs Graphs Number Theory (Coferece hoorig Charles Vade Eyde, Illiois State Uiversity, April 00),