Geodesic
ALTERNATING AND SYMMETRIC GROUPS WITH EULERIAN GENERATING GRAPH
Forum of Mathematics, Sigma, Tome 5 (2017)

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Given a finite group $G$ , the generating graph $\unicode[STIX]{x1D6E4}(G)$ of $G$ has as vertices the (nontrivial) elements of $G$ and two vertices are adjacent if and only if they are distinct and generate $G$ as group elements. In this paper we investigate properties about the degrees of the vertices of $\unicode[STIX]{x1D6E4}(G)$ when $G$ is an alternating group or a symmetric group of degree $n$ . In particular, we determine the vertices of $\unicode[STIX]{x1D6E4}(G)$ having even degree and show that $\unicode[STIX]{x1D6E4}(G)$ is Eulerian if and only if $n\geqslant 3$ and $n$ and $n-1$ are not equal to a prime number congruent to 3 modulo 4. 
@article{10_1017_fms_2017_25,
     author = {ANDREA LUCCHINI and CLAUDE MARION},
     title = {ALTERNATING {AND} {SYMMETRIC} {GROUPS} {WITH} {EULERIAN} {GENERATING} {GRAPH}},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
     volume = {5},
     year = {2017},
     doi = {10.1017/fms.2017.25},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2017.25/}
}
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ANDREA LUCCHINI; CLAUDE MARION. ALTERNATING AND SYMMETRIC GROUPS WITH EULERIAN GENERATING GRAPH. Forum of Mathematics, Sigma, Tome 5 (2017). doi: 10.1017/fms.2017.25

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