Regular handicap tournaments of high degree

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 $n-7$

Distance magic-type and distance antimagic-type labelings of graphs

Generally speaking, a distance magic-type labeling of a graph G of order n is a bijection f from the vertex set of the graph to the first n natural numbers or to the elements of a group of order n, with the property that the weight of each vertex is the same. The weight of a vertex x is defined as the sum (or appropriate group operation) of all the labels of vertices adjacent to x. If instead we require that all weights differ, then we refer to the labeling as a distance antimagic-type labeling. This idea can be generalized for directed graphs; the weight will take into consideration the direction of the arcs. In this manuscript, we provide new results for d-handicap labeling, a distance antimagic-type labeling, and introduce a new distance magic-type labeling called orientable Gamma-distance magic labeling. A d-handicap distance antimagic labeling (or just d-handicap labeling for short) of a graph G=(V,E) of order n is a bijection f from V to {1,2,...,n} with induced weight function w(x_{i})=\underset{x_{j}\in N(x_{i})}{\sum}f(x_{j}) \] such that f(x_{i})=i and the sequence of weights w(x_{1}),w(x_{2}),...,w(x_{n}) forms an arithmetic sequence with constant difference d at least 1. If a graph G admits a d-handicap labeling, we say G is a d-handicap graph. A d-handicap incomplete tournament, H(n,k,d) is an incomplete tournament of n teams ranked with the first n natural numbers such that each team plays exactly k games and the strength of schedule of the ith ranked team is d more than the i+1st ranked team. That is, strength of schedule increases arithmetically with strength of team. Constructing an H(n,k,d) is equivalent to finding a d-handicap labeling of a k-regular graph of order n. In Chapter 2 we provide general constructions for every d at least 1 for large classes of both n and k, providing breadth and depth to the catalog of known H(n,k,d)\u27s. In Chapters 3 - 6, we introduce a new type of labeling called orientable Gamma-distance magic labeling. Let Gamma be an abelian group of order n. If for a graph G=(V,E) of order n there exists an orientation of G and a companion bijection f from V to Gamma with the property that there is an element mu in Gamma (called the magic constant) such that \[ w(x)=\sum_{y\in N_{G}^{+}(x)}\overrightarrow{f}(y)-\sum_{y\in N_{G}^{-}(x)}\overrightarrow{f}(y)=\mu for every x in V where w(x) is the weight of vertex x, we say that G is orientable Gamma-distance magic}. In addition to introducing the concept, we provide numerous results on orientable Z_n distance magic graphs, where Z_n is the cyclic group of order n. In Chapter 7, we summarize the results of this dissertation and provide suggestions for future work

Spartan Daily, October 29, 1942

Volume 31, Issue 21https://scholarworks.sjsu.edu/spartandaily/3501/thumbnail.jp

Graph Labelings and Tournament Scheduling

University of Minnesota M.S. thesis. May 2015. Major: Applied and Computational Mathematics. Advisor: Dalibor Froncek. 1 computer file (PDF); viii, 55 pages.During my research I studied and became familiar with distance magic and distance antimagic labelings and their relation to tournament scheduling. Roughly speaking, the relation is as follows. Let the vertices on the graph represent teams in a tournament, and let an edge between two vertices a and b represent that team a will play team b in the tournament. Further, suppose we can rank the teams based on previous games, say, the preceding season. These integer rankings become labels for the vertices. Of particular interest were handicap tournaments, that is, tournaments designed to give each team a more balanced chance of winning

A note on incomplete regular tournaments with handicap two of order n≡8(mod 16)

A $d$-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 weights $w(x_1),w(x_2),\ldots,w(x_n)$ (where $w(x_i)=\sum_{x_i x_j\in E}f(x_j)$) forms an increasing arithmetic progression with common difference $d$. A graph $G$ is a $d$-handicap distance antimagic graph if it allows a $d$-handicap distance antimagic labeling. We construct a class of $k$-regular $2$-handicap distance antimagic graphs for every order $n\equiv8\pmod{16}$, $n\geq56$ and $6\leq k\leq n-50$

The Observer Vol. 9, Issue No. 12, 04/20/1967

Sandy Johnston Named \u2767 Miss Gorham State -- I.A. Class To Sell Port-A-Grill Product -- Brownell To Address Seniors -- Modern Dancers To Perform April 27, 28 -- Water Tower To Be Repainted, Relettered -- Senate Is Informed Of Budget Procedure By President Brooks -- Bolotowksy Show In Gallery -- Faculty Reminded Not To Smoke In Classhttps://digitalcommons.usm.maine.edu/observer/1040/thumbnail.jp

Неполные турниры и магические типы разметок

Предложен обзор существующих теоретических результатов по вершинным магическим разметкам графов, применяемым в качестве математических моделей в задачах составления расписаний для неполных турниров. Выполнена их систематизация для адаптации к другим видам задач. Методы построения графов неполных турниров разбиты на три группы. Предложены новые подходы для их реализации.Мета — систематизувати основні теоретичні відомості, що стосуються даної тематики, виділити відриті проблеми, класифікувати методи побудови графів турнірів та уніфікувати алгоритми їх опису відповідно до класифікації. Методи. Запропоновано нові алгоритми побудови графів неповних турнірів. Це дає можливість розширити коло задач з використанням математичних моделей на основі розмічених графівPurpose. The purpose of the article is to systematize the main theoretical information related to this topic, to highlight the problems that have not been solved, to classify the methods of constructing graphs of tournaments and to unify the algorithms for their description in accordance with this classification. Methods. New algorithms for constructing incomplete tournaments graphs are offered. This makes it possible to extend the range of tasks using mathematical models based on labeled graphs

Kenyon Collegian - April 30, 1915

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The Tiger Vol. XXVII No. 23 - 1932-03-16

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The Wellesley News (05-13-1908)

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