Magic graph

Magic graphs are in the graph theory , a graph class with special reviews of corners and edges . The weight of an edge is equal to the sum of the evaluations of the beginning and end corners and the edge itself. If all edge weights are the same, one speaks of an edge magic graph . The weight of a corner results from the sum of the corner evaluation and the evaluation of each edge beginning there. If all vertex weights are equal, we speak of vertex magic graphs . Graphs that are both corner and edge magic are called total magic graphs .

Corner magic graphs

Let be a finite simple undirected graph with a total score . resp. are corner-magic, if a corner constant exists, so that for every corner the following applies: ${\displaystyle G=(E,K)}$ ${\displaystyle \lambda :E\cup K\to \{1,2,...,|E|+|K|\}}$ ${\displaystyle G}$ ${\displaystyle \lambda }$ ${\displaystyle h\in \mathbb {N} }$ ${\displaystyle e\in E}$

${\displaystyle wt(e):=\lambda (e)+\sum _{eb\in K}\lambda (eb)=h}$ (Corner weight)

The weight of a corner is the sum of the corner evaluation and the evaluation of each edge beginning there.

Edge magic graphs

Let be a finite simple undirected graph with a total score . or are edge-magic, if an edge constant exists, so that for every edge : ${\displaystyle G=(E,K)}$ ${\displaystyle \lambda :E\cup K\to \{1,2,...,|E|+|K|\}}$ ${\displaystyle G}$ ${\displaystyle \lambda }$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle ab\in K}$

${\displaystyle wt(ab):=\lambda (a)+\lambda (ab)+\lambda (b)=k}$ (Edge weight)

An edge is weighted with the sum of the evaluations of the beginning and end corners and the edge itself.

Totally magical graph

Let be a finite simple undirected graph with a total score . or are totally magical if a corner constant and an edge constant exist, so that or is both corner and edge magic. ${\displaystyle G=(E,K)}$ ${\displaystyle \lambda :E\cup K\to \{1,2,...,|E|+|K|\}}$ ${\displaystyle G}$ ${\displaystyle \lambda }$ ${\displaystyle h\in \mathbb {N} }$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle G}$ ${\displaystyle \lambda }$

Examples

• The trivial graph (graph with a corner and no edge) is totally magical with the corner constant . The edge constant is debatable. ${\displaystyle K_{1}}$ ${\displaystyle 1}$
• The circle graph (triangle) is totally magical. ${\displaystyle C_{3}}$
• The linear graph is totally magical. ${\displaystyle P_{3}}$
• ${\displaystyle P_{3}}$and are the only totally magical stars . ${\displaystyle K_{1}}$
• The graph is totally magical. ${\displaystyle K_{1}\cup P_{3}}$

literature

• Alison M. Marr, WD Wallis: Magic Graphs . 2nd Edition. Springer, 2012, ISBN 978-0-8176-8391-7 .
• A. Kotzig, A. Rosa: Magic valuations of finite graphs . In: Canad. Math. Bull. , 13, 1970, pp. 451-461

Web links

• Eric W. Weisstein : Magic graph . In: MathWorld (English).
• Totally magic graphs