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| In [[graph theory]], an '''edge-graceful''' graph labeling is a type of [[graph labeling]]. This is a labeling for [[simple graph]]s in which no two distinct [[edge (graph theory)|edge]]s connect the same two distinct [[Vertex (graph theory)|vertices]], no edge connects a vertex to itself, and the graph is [[Connected graph|connected]]. Edge-graceful labelings were first introduced by S. Lo in his seminal paper.<ref>Lo (1985)</ref>
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| == Definition ==
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| Given a graph ''G'', we denote the set of edges by ''E''(''G'') and the vertices by ''V''(''G''). Let q be the [[cardinality]] of ''E''(''G'') and ''p'' be that of ''V''(''G''). Once a labeling of the edges is given, a vertex ''u'' of the graph is labeled by the sum of the labels of the edges incident to it, modulo ''p''. Or, in symbols, the induced labeling on the vertex ''u'' is given by
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| :<math>V(u)=\Sigma E(e) \mod |V(G)|</math>
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| where ''V''(''u'') is the label for the vertex and ''E''(''e'') is the assigned value of an edge incident to ''u''.
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| The problem is to find a labeling for the edges such that all the labels from 1 to ''q'' are used once and the induced labels on the vertices run from 0 to ''p'' − 1. In other words, the resulting set for labels of the edges should be <math>\{1,2 \dots q\}</math> and <math>\{0,1 \dots p-1\}</math> for the vertices. | |
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| A graph ''G'' is said to be edge-graceful if it admits an edge-graceful labeling.
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| == Examples ==
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| === Paths ===
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| Consider a [[path (graph theory)|path]] with two vertices, ''P''<sub>2</sub>. Here the only possibility is to label the only edge in the graph 1. The induced labeling on the two vertices are both 1. So ''P''<sub>2</sub> is not edge-graceful.
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| Appending an edge and a vertex to ''P''<sub>2</sub> gives ''P''<sub>3</sub>, the path with three vertices. Denote the vertices by ''v''<sub>1</sub>, ''v''<sub>2</sub>, and ''v''<sub>3</sub>. Label the two edges in the following way: the edge (''v''<sub>1</sub>, ''v''<sub>2</sub>) is labeled 1 and (''v''<sub>2</sub>, ''v''<sub>3</sub>) labeled 2. The induced labelings on ''v''<sub>1</sub>, ''v''<sub>2</sub>, and ''v''<sub>3</sub> are then 1, 0, and 2 respectively. This is an edge-graceful labeling and so ''P''<sub>3</sub> is edge-graceful.
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| Similarly, one can check that ''P''<sub>4</sub> is not edge-graceful.
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| In general, ''P''<sub>m</sub> is edge-graceful when ''m'' is odd and not edge-graceful when it is even. This follows from a necessary condition for edge-gracefulness (see below).
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| === Cycles ===
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| Consider the [[cycle graph|cycle]] with three vertices, ''C''<sub>3</sub>. This is simply a triangle. One can label the edges 1, 2, and 3, and check directly that, along with the induced labeling on the vertices, this gives an edge-graceful labeling.
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| Similar to paths, <math>C_m</math> is edge-graceful when ''m'' is odd and not when ''m'' is even.
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| An edge-graceful labeling of <math>C_5</math> is shown in the following figure:
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| [[Image:Edge graceful c5.svg]] | |
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| == A necessary condition ==
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| Lo gave a necessary condition for a graph to be edge-graceful. It is that a graph with ''q'' edges and ''p'' vertices is edge graceful only if
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| :<math>\; q(q+1)</math> is [[Congruence relation#modular arithmetic|congruent]] to <math>\frac{p(p-1)}{2}</math> modulo ''p''.
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| or, in symbols,
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| :<math>q(q+1) \equiv \frac{p(p-1)}{2} \mod p.</math>
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| This is referred to as '''Lo's condition''' in the literature. This follows from the fact that the sum of the labels of the vertices is twice the sum of the edges, modulo ''p''. This is useful for disproving a graph is edge-graceful. For instance, one can apply this directly to the path and cycle examples given above.
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| == Further selected results ==
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| *The [[Petersen graph]] is not edge-graceful.
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| *The [[star graph]] <math>S_m</math> (a central node and ''m'' legs of length 1) is edge-graceful when ''m'' is [[even number|even]] and not when ''m'' is [[odd number|odd]].
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| *The [[friendship graph]] <math>F_m</math> is edge-graceful when ''m'' is odd and not when it is even.
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| *[[Regular tree]]s, <math>T_{m,n}</math> (depth ''n'' with each non-leaf node emitting ''m'' new vertices) are edge-graceful when ''m'' is even for any value ''n'' but not edge-graceful whenever ''m'' is odd.
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| *The [[complete graph]] on ''n'' vertices, <math>K_n</math>, is edge-graceful unless ''n'' is [[singly even]], <math>n=2\mod 4</math>.
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| *The [[ladder graph]] is never edge-graceful.
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| == References ==
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| {{reflist}}
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| * {{cite journal | zbl=0597.05054 |last=Lo | first=Sheng-Ping | title=On edge-graceful labelings of graphs |
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| conference=Proc. Conf., Sundance/Utah 1985 | journal=Congressus Numerantium | volume=50 | year=1985 | pages=231–241 }}
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| * Q. Kuan, S. Lee, J. Mitchem, and A. Wang, ''On Edge-Graceful Unicyclic Graphs'', Congressus Numerantium 61(1988) pp. 65–74
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| * L. Lee, S. Lee and G. Murty, ''On Edge-Graceful Labelings of Complete Graphs: Solutions of Lo’s Conjecture'', Congressus Numerantium 62(1988) pp. 225–233
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| * D. Small, ''Regular (even) Spider Graphs are Edge-Graceful'', Congressus Numerantium 74(1990) pp. 247–254
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| * S. Cabaniss, R. Low, J. Mitchem, ''On Edge-Graceful Regular Graphs and Trees'', Ars Combinatoria 34(1992) pp. 129–142
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| {{DEFAULTSORT:Edge-Graceful Labeling}}
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| [[Category:Graph theory]]
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