A graph has a representation modulo n if there exists an injective map f: {V(G)} -\u3e {0, 1, ... , n @ 1} such that vertices u and v are adjacent if and only if |f(u)@f(v)| is relatively prime to n. The representation number is the smallest n such that G has a representation modulo n. We seek the maximum value for the representation number over graphs of a fixed order. Erdos and Evans provided an upper bound in their proof that every finite graph can be represented modulo some positive integer. In this note we improve this bound and show that the new bound is best possible (Refer to PDF file for exact formulas)

Narayan, Darren

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An Upper Bound for the Representation Number of Graphs with Fixed Order

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Rochester Institute of TechnologyRIT Scholar WorksArticles2003An upper bound for the representation number ofgraphs with fixed orderDarren NarayanFollow this and additional works at: http://scholarworks.rit.edu/articleThis Article is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Articles by an authorizedadministrator of RIT Scholar Works. For more information, please contact ritscholarworks@rit.edu.Recommended CitationNarayan, Darren, "An upper bound for the representation number of graphs with fixed order" (2003). Integers: Electronic Journal ofCombinatorial Number Theory, vol. 3 (), A12. Accessed fromhttp://scholarworks.rit.edu/article/289AN UPPER BOUND FOR THE REPRESENTATION NUMBER OFGRAPHS WITH FIXED ORDERDarren A. NarayanDepartment of Mathematics and Statistics, Rochester Institute of Technology, Rochester, NY 14623dansma@rit.eduReceived: 4/30/03, Revised: 7/17/03, Accepted: 7/31/03, Published: 8/7/03AbstractA graph has a representation modulo n if there exists an injective map f : {V (G)} →{0; 1; : : : ; n − 1} such that vertices u and v are adjacent if and only if |f(u)− f(v)| isrelatively prime to n. The representation number is the smallest n such that G has arepresentation modulo n. We seek the maximum value for the representation numberover graphs of a ¯xed order. Erd}os and Evans provided an upper bound in their proofthat every ¯nite graph can be represented modulo some positive integer. In this notewe improve this bound and show that the new bound is best possible.1. IntroductionLet G be a ¯nite graph with vertices {v1; : : : ; vr}. A representation of G modulo n is anassignment of distinct labels to the vertices such that the label ai assigned to vertex viis in {0; 1; : : : ; n − 1} and such that |ai − aj| and n are relatively prime if and only if(vi; vj) ∈ E(G). Erd}os and Evans [1] showed that every ¯nite graph can be representedmodulo some positive integer. The representation number of a graph G, denoted rep (G);is the smallest n such that G has a representation modulo n.Modular representations have received considerable attention in recent years as asource of open problems (see [4] and [7]). Representation numbers for various classes ofgraphs were determined in [2] and [3], but little is known for many families of graphs,including bipartite graphs and trees.The existence proof in [1] is elegant but gives an unnecessarily large upper bound forthe representation number. For a graph of order r, the value n is the product of r primes,each greater than 3(r2). Our bound is the product of the ¯rst r − 1 primes greater thanor equal to r − 1. In fact we show that this signi¯cantly smaller bound is best possibleover all graphs of order r.INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 3 (2003), #A12 2We also mention a connection to a result involving orthogonal latin square graphs.An orthogonal latin square graph is one whose vertices can be labeled with latin squaresof the same order and same symbols such that two vertices are adjacent if and only if thecorresponding latin squares are orthogonal. Lindner, E. Mendelsohn, N. S. Mendelsohn,and Wolk [5] showed that every ¯nite graph is an orthogonal latin square graph. Ashorter proof of this result was given by Erd}os and Evans [1] after establishing that every¯nite graph can be represented modulo some positive integer. An even more simple proofof the theorem from [5] can be obtained using the upper bound found in this note.2. Dimensions and representationsA product representation of length t assigns distinct vectors of length t to each vertexso that vertices u and v are adjacent if and only if their vectors di®er in every position.The product dimension of a graph, denoted pdimG, is the minimum length of such arepresentation of G.As developed in [2] and [3], there is a close correspondence between product repre-sentation and modular representation. From a representation of a graph G modulo aproduct of primes q1; : : : ; qt, we obtain a product representation of length t as follows.The vector for vertex v is (v1; : : : ; vt), where vi ≡ a(mod qi) and vi ∈ {0; : : : ; qi−1} for1 ≤ i ≤ t. If u has vector (u1; : : : ; ut) and v has vector (v1; : : : ; vt), then the modularrepresentation implies that u and v are adjacent if and only if ui 6= vi for all i, makingthis assignment a product representation.Conversely, given a product representation, a modular representation can be obtainedby choosing distinct primes for the coordinates, provided that the prime for each coordi-nate is larger than the number of values used in that coordinate. The numbers assignedto the vertices can then be obtained using the Chinese Remainder Theorem. The re-sulting modular representation generated from the product representation is called thecoordinate representation.We use pi to denote the ith prime, and for any prime pi we use pi+1; pi+2; : : : ; pi+k todenote the next k primes larger than pi. The seminal work on product dimension wasdone by Lov¶asz, Ne·set·ril, and Pultr [6]. We ¯rst restate one of their results and then aresult from [3] as Lemmas 1 and 2. The graph Kr−1 + K1 is the disjoint union of Kr−1and K1.Lemma 1 For r ≥ 3; the maximum product dimension of an r−vertex graph is r − 1,achieved by Kr−1 +K1.Lemma 2 Let ps be the smallest prime that is at least r − 1. Then rep (Kr−1 + K1) =psps+1 · · · ps+r−2.INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 3 (2003), #A12 3By converting from product representations to modular representations, we showthat for all r ≥ 3; Kr−1 + K1 is the r-vertex graph with largest representation number.For graphs with at most two vertices, it is straightforward to show that rep (K1) = 1,rep (K2) = 2, and rep (2K1) = 4.Theorem 3 For r ≥ 3, the maximum of rep (G) over graphs of order r is psps+1 · · · ps+r−2,where ps is the smallest prime that is at least r − 1.Proof. The sharpness of the upper bound follows from Lemma 2. To prove the upperbound, let G have order r, and begin with a product representation of length r − 1provided by Lemma 1. By relabeling if necessary, we may assume that the values usedin coordinate i are {0; 1; : : : ; ci− 1} for some positive integer ci and that coordinates areindexed such that c1 ≤ · · · ≤ cr−1.If G is not complete, then c1 ≤ r − 1. Thus we may associate ps+i−1 with theith coordinate and the corresponding coordinate representation of G is a representationmodulo ps · · · ps+r−2. If G is complete, then rep (G) is the smallest prime that is at leastr, which is smaller than the claimed upper bound.3. ConclusionWe note that the construction given in the proof of Theorem 3 will not always givethe representation number, since the representation number need not be a product ofdistinct primes. The case mentioned earlier, rep(2K1) = 4, is one of in¯nitely manyexamples. Therefore, ¯nding the representation number of a graph is a di®erent problemfrom ¯nding the product dimension of a graph. Since the representation number of agraph depends upon the distribution of primes and prime powers, tools from numbertheory may be valuable for future studies. There are many open problems involvingmodular repesentations. Representation numbers have been determined for only a fewgraph families (see [2] and [3]). Little is known about representation numbers for somemultipartite graphs, including the most basic cases involving trees and complete bipartitegraphs.AcknowledgementsThe author is indebted to a referee for many valuable suggestions, including an im-proved presentation of the proof of Theorem 3. This work was partially supported by aRIT College of Science Dean's Summer Research Fellowship Grant and a Reidler Founda-tion Grant. Thanks to Anthony Evans and Garth Isaak for useful discussions. Finally theauthor thanks Dustin She±eld, now an undergraduate at Virginia Tech, for preliminarycomputer simulations.INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 3 (2003), #A12 4References[1] P. Erd}os and A. B. Evans, Representations of graphs and orthogonal latin square graphs, J. GraphTheory 13 (1989) 593-595.[2] A. B. Evans, G. H. Fricke, C. C. Maneri, T. A. McKee and M. Perkel, Representations of graphsmodulo n; J. Graph Theory 18 (1994) 801-815.[3] A. B. Evans, G. Isaak and D. A. Narayan, Representations of graphs modulo n: Discrete Math 223(2000) 109-123.[4] J. A. Gallian, A dynamic survey of graph labeling, Electronic. J. Combin. 5 (2002), 1-106.[5] C. C. Lindner, E. Mendelsohn, N. S. Mendelsohn, and B. Wolk, Orthogonal latin square graphs, J.Graph Theory 3 (1979) 325-338.[6] L. Lov¶asz, J. Ne·set·ril, and A. Pultr. On a product dimension of graphs, J. Combin. Theory B 29(1980), 47-67.[7] D. B. West, (ed.) Research Problems, Discrete Math 257 (2002) 599-624.

A graph has a representation modulo n if there exists an injective map f: {V (G)} → {0, 1,...,n − 1} such that vertices u and v are adjacent if and only if |f(u) − f(v)| is relatively prime to n. The representation number is the smallest n such that G has a representation modulo n. We seek the maximum value for the representation number over graphs of a fixed order. Erd˝os and Evans provided an upper bound in their proof that every finite graph can be represented modulo some positive integer. In this note we improve this bound and show that the new bound is best possible

Narayan, Darren A.

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

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An Upper Bound for the Representation Number of Graphs with Fixed Order

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