A handicap distance antimagic labeling of a graph $G=(V,E)$ with $n$ vertices is a bijection ${f}: V\to \{ 1,2,\ldots ,n\} $ with the property that ${f}(x_i)=i$ and the sequence of the weights $w(x_1),w(x_2),\ldots,w(x_n)$ (where $w(x_i)=\sum\limits_{x_j\in N(x_i)}f(x_j)$) forms an increasing arithmetic progression with difference one. A graph $G$ is a {\em handicap distance antimagic graph} if it allows a handicap distance antimagic labeling. We construct $(n-7)$-regular handicap distance antimagic graphs for every order $n\equiv2\pmod4$ with a few small exceptions. This result complements results by Kov\'a\v{r}, Kov\'a\v{r}ov\'a, and Krajc~[P. Kov\'a\v{r}, T. Kov\'a\v{r}ov\'a, B. Krajc, On handicap labeling of regular graphs, manuscript, personal communication, 2016] who found such graphs with regularities smaller than $n-7$

Froncek, Dalibor

Shepanik, Aaron

Journal of Algebra Combinatorics Discrete Structures and Applications

English

A  handicap distance antimagic labeling of a graph $G=(V,E)$ with $n$ vertices is a bijection ${f}: V\to \{ 1,2,\ldots ,n\} $ with the property that ${f}(x_i)=i$ and the sequence of the weights $w(x_1),w(x_2),\ldots,w(x_n)$ (where $w(x_i)=\sum\limits_{x_j\in N(x_i)}f(x_j)$) forms an increasing arithmetic progression with difference one. A graph $G$ is a {\em handicap distance antimagic graph} if it allows a handicap distance antimagic labeling.  We construct $(n-7)$-regular handicap distance antimagic graphs for every order $n\equiv2\pmod4$ with a few small exceptions. This result complements results by Kov\'a\v{r}, Kov\'a\v{r}ov\'a, and Krajc~[P. Kov\'a\v{r}, T. Kov\'a\v{r}ov\'a, B. Krajc, On handicap labeling of regular graphs, manuscript, personal communication, 2016] who found such graphs with regularities smaller than $n-7$.</p

Dalibor Froncek

Aaron Shepanik

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Regular handicap tournaments of high degree

Journal of Algebra Combinatorics Discrete Structures and Applications (JACODESMATH, Yildiz Technical University - YTU)

ISSN 2148-838Xhttp://dx.doi.org/10.13069/jacodesmath.22530J. Algebra Comb. Discrete Appl.3(3) • 159–164Received: 07 December 2015Accepted: 19 March 2016Journal of Algebra Combinatorics Discrete Structures and ApplicationsRegular handicap tournaments of high degreeResearch ArticleDalibor Froncek, Aaron ShepanikAbstract: A handicap distance antimagic labeling of a graph G = (V,E) with n vertices is a bijection f : V →{1, 2, . . . , n} with the property that f(xi) = i and the sequence of the weights w(x1), w(x2), . . . , w(xn)(where w(xi) =∑xj∈N(xi)f(xj)) forms an increasing arithmetic progression with difference one. Agraph G is a handicap distance antimagic graph if it allows a handicap distance antimagic labeling.We construct (n− 7)-regular handicap distance antimagic graphs for every order n ≡ 2 (mod 4) witha few small exceptions. This result complements results by Kovář, Kovářová, and Krajc [P. Kovář, T.Kovářová, B. Krajc, On handicap labeling of regular graphs, manuscript, personal communication,2016] who found such graphs with regularities smaller than n− 7.2010 MSC: 05C78Keywords: Incomplete tournaments, Handicap tournaments, Distance magic labeling, Handicap labeling1. MotivationThe study of handicap distance antimagic graphs has been motivated by incomplete round-robintype tournaments with various properties.A complete round robin tournament of n teams is a tournament in which every team plays theremaining n− 1 teams. When the teams are ranked 1, 2, . . . , n according to their strength, it is apparentthat the sum of rankings of all opponents of the i-th ranked team, denoted w(i), is w(i) = n(n+ 1)/2− i,and the sequence w(1), w(2), . . . , w(n) is a decreasing arithmetic progression with difference one. Becausecomplete round robin tournaments are generally considered to be fair, a tournament of n teams in whichevery team plays precisely r opponents, where r < n − 1 and the sequence w(1), w(2), . . . , w(n) is adecreasing arithmetic progression with difference one is called a fair incomplete round robin tournament.A disadvantage of such a tournament is that the best team plays the weakest opponents, while the weakestteam plays the strongest opponents. This disadvantage is eliminated in equalized incomplete round robintournaments in which the sum of rankings of all opponents of every team is the same. Some results onfair incomplete round robin tournaments can be found in [6] and [3].Dalibor Froncek (Corresponding Author), Aaron Shepanik; Department of Mathematics and Statistics, Universityof Minnesota Duluth, USA (email: dfroncek@d.umn.edu, shepa107@d.umn.edu).159D. Froncek, A. Shepanik / J. Algebra Comb. Discrete Appl. 3(3) (2016) 159–164However, if we want to give the weaker teams a better chance of winning, the weakest team shouldplay the weakest opponents, while the strongest one should play the strongest opponents. That is, thesequence w(1), w(2), . . . , w(n) should be an increasing arithmetic progression. A tournament in whichthis condition is satisfied, and every team plays r < n− 1 games, is called a handicap incomplete roundrobin tournament.The existence of such tournaments with n ≡ 0 (mod 4) is studied by the authors in [D. Froncekand A. Shepanik, Handicap incomplete tournaments of order n ≡ 0 (mod 4), manuscript, personal com-munication, 2016], Kovář [P. Kovář, On regular handicap graphs, personal communication, June 16,2016] and Kovářová [T. Kovářová, On regular handicap graphs, personal communication, June 16, 2016].Kovář, Kovářová, and Krajc [P. Kovář, T. Kovářová, B. Krajc, On handicap labeling of regular graphs,manuscript, personal communication, 2016] found such tournaments for n ≡ 2 (mod 4) and r ≤ n − 11and proved that they can exist only when r is odd and at most n − 7. We provide a construction ofhandicap incomplete round robin tournaments for n ≡ 2 (mod 4) and the missing regularity r = n − 7with a few small exceptions.2. Basic notionsBy a graph G = (V,E) we mean a finite undirected graph without loops or multiple edges. For graphtheoretic terminology we refer to Chartrand and Lesniak [2].Motivated by properties of magic squares, Vilfred [12] introduced the concept of sigma labelings. Thesame concept was introduced by Miller et al. [10] under the name 1-vertex magic vertex labeling. Sugenget al. [11] introduced the term distance magic labeling, which currently seems to be most commonlyused. A survey on distance magic graphs was published recently [1]. Many newer results can by foundin an extensive survey with much wider focus by Gallian [7].Definition 2.1. A distance magic labeling of a graph G of order n is a bijection f : V → {1, 2, . . . , n}with the property that there is a positive integer µ such that∑y∈N(x)f(y) = µ for every x ∈ V. The constantµ is called the magic constant of the labeling f. The sum∑y∈N(x)f(y) is called the weight of the vertex xand is denoted by w(x).When we think of the vertices as of teams and identify their labels with their rankings, we can seethat a distance magic graph is providing a structure of a fair incomplete tournament described above.In [4] the first author introduced two closely related concepts, namely the distance antimagic andhandicap distance antimagic labelings (which was called an ordered distance antimagic labeling in thatpaper) and showed their relationship to certain types of incomplete round robin tournaments. Theterm “handicap distance antimagic labeling” was originally coined by Kovářová [T. Kovářová, On regularhandicap graphs, personal communication, June 16, 2016].Definition 2.2. A distance d-antimagic labeling of a graph G = (V,E) with n vertices is a bijectionf : V → {1, 2, . . . , n} with the property that there exists an ordering of the vertices of G such that thesequence of the weights w(x1), w(x2), . . . , w(xn) forms an arithmetic progression with difference d. Whend = 1, then f is called just distance antimagic labeling. A graph G is a distance d-antimagic graph if itallows a distance d-antimagic labeling, and a distance antimagic graph when d = 1.It should be obvious that a graph G is distance magic if and only if its complement G is distanceantimagic.In distance antimagic graphs the weight of a vertex is not tied to its own label. All that we require isthat the sequence w(x1), w(x2), . . . , w(xn) forms an arithmetic progression. We now impose an additionalcondition on the labeling and require that a vertex with a lower label has a lower weight than a vertexwith a higher label.160D. Froncek, A. Shepanik / J. Algebra Comb. Discrete Appl. 3(3) (2016) 159–164Definition 2.3. A handicap distance d-antimagic labeling of a graph G = (V,E) with n vertices isa bijection f : V → {1, 2, . . . , n} with the property that f(xi) = i and the sequence of the weightsw(x1), w(x2), . . . , w(xn) forms an increasing arithmetic progression with difference d. When d = 1, thelabeling is called just a handicap distance antimagic labeling (or a handicap labeling for short).A graph G is a handicap distance d-antimagic graph if it allows a handicap distance d-antimagiclabeling, and a handicap distance antimagic graph or a handicap graph when d = 1.Again, if we identify each team in a tournament with its ranking, then an r-regular handicap distanced-antimagic graph is nothing else than a model of a handicap incomplete round robin tournament, sincethe sum of rankings of opponents of team i is its weight w(i) and the sequence of weights is an increasingarithmetic progression.Our constructions will be based on the properties of magic rectangles, which are a generalization ofthe magic squares mentioned above.Definition 2.4. A magic rectangle MR(a, b) is an a × b array whose entries are 1, 2, . . . , ab, each ap-pearing once, with all row sums equal to a constant ρ and all column sums equal to a constant σ.It is easy to observe that a and b must be either both even or both odd. The following existenceresult was proved by Harmuth [8, 9] more than 130 years ago.Theorem 2.5. [8, 9] A magic rectangle MR(a, b) exists if and only if a, b > 1, ab > 4, and a ≡ b(mod 2).3. Known resultsKovář, Kovářová, and Krajc [P. Kovář, T. Kovářová, B. Krajc, On handicap labeling of regulargraphs, manuscript, personal communication, 2016], Kovář [P. Kovář, On regular handicap graphs, per-sonal communication, June 16, 2016] and Kovářová [T. Kovářová, On regular handicap graphs, personalcommunication, June 16, 2016] proved the following results.Theorem 3.1. Let G be an r-regular handicap graph on n vertices, where n ≡ 2 (mod 4). Then r ≡ 3(mod 4) and r ≤ n− 7.Theorem 3.2. For n ≡ 2 (mod 4), there exists an r-regular handicap graph if 3 ≤ r ≤ n− 11 and r ≡ 3(mod 4) except when r = 3 and n ≤ 26.Their result leaves open the case of n ≡ 2 (mod 4) and r = n−7 for n ≥ 14. In the following section,we prove the existence of such graphs with the exception of n = 14, 18, 22, 26, 34, 38, which remain indoubt.In our constructions, we will also use the following result by Kovář [P. Kovář, On regular handicapgraphs, personal communication, June 16, 2016] and Kovářová [T. Kovářová, On regular handicap graphs,personal communication, June 16, 2016].Theorem 3.3. For n ≡ 0 (mod 4) there exists an (n− 7)-regular handicap graph whenever n ≥ 16.In [5] the first author made made an observation, which was a special case of the following.Observation 3.4. Let G be an r-regular distance 2-antimagic graph with vertices x1, x2, . . . , xn, labelingf and weight function w such that f(xi) = i and w(xi) = k − 2i for some constant k. Then G, thecomplement of G, is an (n − r − 1)-regular handicap graph with labeling f and weight function w suchthat w(xi) = n(n+ 1)/2− k + i. The converse is obviously also true.Proof. Label the vertices of the complete graph Kn so that vertex xi is labelled i. The sum of labelsof all neighbors of xi is then indeed equal to n(n+ 1)/2− i. Every neighbor of xi contributes its label to161D. Froncek, A. Shepanik / J. Algebra Comb. Discrete Appl. 3(3) (2016) 159–164either w(xi) or w(xi). Therefore, we havew(xi) + w(xi) = n(n+ 1)/2− iandw(xi) = n(n+ 1)/2− i− w(xi).Because w(xi) = k − 2i, it follows thatw(xi) = n(n+ 1)/2− i− (k − 2i) = n(n+ 1)/2− k + i.This completes the proof.We will use the observation in our constructions and instead of constructing directly (n− 7)-regularhandicap graphs, we will construct 6-regular distance 2-antimagic graphs satisfying assumptions of Ob-servation 3.4. We will call such graphs genuine distance 2-antimagic graphs and the labeling will be calleda genuine distance 2-antimagic labeling.The following observation was proved in a more general form in [4]. To avoid introduction of a newnotion that would be only used in its special form, we state the observation as follows.Observation 3.5. [4] The graph G = KaKb admits a genuine distance 2-antimagic labeling f suchthat f(x) = p implies w(x) = (a + b)(ab + 1)/2 − 2p for every x ∈ V (G) whenever there exists a magicrectangle MR(a, b).4. New resultsLemma 4.1. There exists a 6-regular genuine distance 2-antimagic graph on 12 vertices.Proof. Let G = K2K6. By Theorem 2.5 there exist a magic rectangle MR(2, 6). The assertionfollows directly from Observation 3.5.Lemma 4.2. There exists a 6-regular genuine distance 2-antimagic graph H on 30 vertices.Proof. We denote the vertices of H by yst and zst for 1 ≤ s ≤ 3 and 1 ≤ t ≤ 5. The edge set willconsist of edges ystypt, zstzpt, and ystzsq for every for every 1 ≤ s ≤ p ≤ 3 and 1 ≤ t ≤ 5 with t 6= q.Let MR(3, 5) be a 3 × 5 magic rectangle with entries mst for 1 ≤ r ≤ 3 and 1 ≤ s ≤ 5. We labelthe vertices yst by entries mst in the natural way, that is, f(yst) = mst while the vertices zst obtain thelabels raised by 15, that is, f(zst) = mst + 15. Notice that the vertices yst have the highest weights.We now rename the vertices so that vertex yst or zst becomes xi when w(yst) = 124− 2i or w(zst) =124 − 2i, respectively. One can check that this labeling has the required property and H is a 6-regulargenuine distance 2-antimagic graph.Now we are ready to present our construction.Lemma 4.3. There exists a 6-regular genuine distance 2-antimagic graph G on n vertices for everyn ≡ 2 (mod 4) and n ≥ 42.Proof. We use two building blocks, graph H on 30 vertices from the previous Lemma, and a graph Jon m ≡ 0 (mod 4) vertices whose existence is guaranteed by Lemma 4.1 and Theorem 3.3. Our graph Gwill then have n = m+ 30 vertices.Let H be the graph on 30 vertices constructed in Lemma 4.2 with vertices x1, x2, . . . , x30, genuinedistance 2-antimagic labeling fH and vertex weights wH satisfying fH(xi) = i and wH(xi) = 124− 2i.162D. Froncek, A. Shepanik / J. Algebra Comb. Discrete Appl. 3(3) (2016) 159–164It follows from Theorem 3.3 and Observation 3.4 that for any m ≡ 0 (mod 4) there exists a genuinedistance 2-antimagic labeling graph J with vertices u1, u2, . . . , um, labeling fJ and vertex weights wJsatisfying fJ(uj) = j and wJ(uj) = 4m− 2j.We useH and J as components of G and rename the vertices so that xi becomes vi for i = 1, 2, . . . , 15,uj becomes vj+15 for j = 1, 2, . . . ,m and xi becomes vi+m for i = 16, 17, . . . , 30. Then we label all verticesusing labeling function fG as fG(vi) = i to obtain the desired genuine distance 2-antimagic labeling.We can check that for i = 1, 2, . . . , 15 we havewG(vi) = wH(xi) + 4m = 124− 2i+ 4m = 4(m+ 30) + 4− 2i = 4n+ 4− 2i,because vi has two neighbors in the “lower” part of H (that is, among vertices v1, v2, . . . , v15) where thelabels have not changed, and four neighbors in the “upper” part (among vertices vm+16, vm+17, . . . , vm+30),where each label was increased by m. For i = 16, 17, . . . ,m+ 15 we havewG(vi) = wJ(ui−15) + 90 = 4m+ 4− 2(i− 15) + 90 = 4(m+ 30) + 4− 2i = 4n+ 4− 2i,because the labels of all six neighbors were increased by 15. Finally, for i = m+ 16,m+ 17, . . . ,m+ 30we havewG(vi) = wH(xi−m) + 2m = 124− 2(i−m) + 2m = 4(m+ 30) + 4− 2i = 4n+ 4− 2i,because vi has four neighbors in the “lower” part of H where the labels have not changed, and twoneighbors in the “upper” part where each label was increased by m.Our main result now follows directly from Lemma 4.3 and Observation 3.4.Theorem 4.4. For n ≡ 2 (mod 4), there exists an (n − 7)-regular handicap graph when n = 30 orn ≥ 42.This, together with the result by Kovář [P. Kovář, On regular handicap graphs, personal communi-cation, June 16, 2016] and Kovářová [T. Kovářová, On regular handicap graphs, personal communication,June 16, 2016], gives an almost complete characterization of handicap graphs for n ≡ 2 (mod 4).Theorem 4.5. For n ≡ 2 (mod 4), there exists an r-regular handicap graph if and only if 3 ≤ r ≤n− 7 and r ≡ 3 (mod 4) except when r = 3 and n ≤ 26 and possibly when r = n − 7 and n ∈{14, 18, 22, 26, 34, 38}.Since no magic rectangles of orders 14, 22, 26, 34, or 38 exist, one has to hope that a computer aidedsearch would help to settle the existence question for these orders. A construction for n = 18 similar tothat for n = 30 may exist, but the authors were unable to find one.References[1] S. Arumugam, D. Froncek, N. Kamatchi, Distance magic graphs – a survey, J. Indones. Math. Soc.Special Edition (2011) 1–9.[2] G. Chartrand, L. Lesniak, Graphs and Digraphs, Chapman and Hall, CRC, Fourth edition, 2005.[3] D. Froncek, Fair incomplete tournaments with odd number of teams and large number of games,Congr. Numer. 187 (2007) 83–89.[4] D. Froncek, Handicap distance antimagic graphs and incomplete tournaments, AKCE Int. J. GraphsComb. 10(2) (2013) 119–127.[5] D. Froncek, Handicap incomplete tournaments and ordered distance antimagic graphs, Congr. Nu-mer. 217 (2013) 93–99.163D. Froncek, A. Shepanik / J. Algebra Comb. Discrete Appl. 3(3) (2016) 159–164[6] D. Froncek, P. Kovář, T. Kovářová, Fair incomplete tournaments, Bull. Inst. Combin. Appl. 48 (2006)31–33.[7] J. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. 5 (# DS6) (2015) 43 pp.[8] T. Harmuth, Ueber magische Quadrate und ähnliche Zahlenfiguren, Arch. Math. Phys. 66 (1881)286–313.[9] T. Harmuth, Ueber magische Rechtecke mit ungeraden Seitenzahlen, Arch. Math. Phys. 66 (1881)413–447.[10] M. Miller, C. Rodger, R. Simanjuntak, Distance magic labelings of graphs, Australas. J. Combin. 28(2003) 305–315.[11] K. A. Sugeng, D. Froncek, M. Miller, J. Ryan, J. Walker, On distance magic labeling of graphs, J.Combin. Math. Combin. Comput. 71 (2009) 39–48.[12] V. Vilfred, Σ-labelled graph and circulant graphs, Ph.D. Thesis, University of Kerala, Trivandrum,India, 1994.164

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Regular handicap tournaments of high degree

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