By Jack Koolen, Jin Ho Kwak, Ming-Yao Xu

**Applications of workforce thought to Combinatorics** includes eleven survey papers from foreign specialists in combinatorics, crew conception and combinatorial topology. The contributions disguise themes from relatively a various spectrum, similar to layout concept, Belyi services, team conception, transitive graphs, ordinary maps, and Hurwitz difficulties, and current the state of the art in those parts. **Applications of team idea to Combinatorics** can be invaluable within the research of graphs, maps and polytopes having maximal symmetry, and is geared toward researchers within the components of team idea and combinatorics, graduate scholars in arithmetic, and different experts who use team concept and combinatorics.

**Jack Koolen** teaches on the division of arithmetic at Pohang college of technology and know-how, Korea. His major examine pursuits comprise the interplay of geometry, linear algebra and combinatorics, on which he released 60 papers.

**Jin Ho Kwak** is Professor on the division of arithmetic at Pohang collage of technology and know-how, Korea, the place he's director of the Combinatorial and Computational arithmetic middle (Com2MaC). He works on combinatorial topology, commonly on masking enumeration with regards to Hurwitz difficulties and average maps on surfaces, and released greater than a hundred papers in those areas.

**Ming-Yao Xu** is Professor in division of arithmetic at Peking collage, China. the point of interest in his study is in finite staff concept and algebraic graph idea. Ming-Yao Xu released over eighty papers on those topics.

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Xu, s-Regular cubic Cayley graphs on abelian or dihedral groups, Institute of Mathematics and School of Mathematical Sciences, Research Report No. 53, 2000. -Q. H. Y. -X. Zhou, Tetravalent half-arc-transitive graphs of order p4 , Europ. J. , 29(2008), 555–567. -Q. J. L. Chen, A family of nonnormal Cayley digraphs, Acta Mathematica Sinica, English Series, 17(2001), 147–152. -Q. J. Y. Xu, Automorphism groups of 2-valent connected Cayley digraphs on regular p-groups, Graphs and Combinatorics, 18(2002), 253–257.

Let ( , P ) be a G-vertex-symmetrical decomposition of a connected graph G intransitive on edges. 9. with Proof. Let E1 , . . , Er be the orbits of G on E . Since G is vertex-transitive, each [Ei ] is a spanning subgraph of . Let P = {P1 , . . , Pk } and for each i ∈ {1, . . , r} and s ∈ {1, . . , k} let Qis = Ei ∩Ps . Then for i ∈ {1, . . , r}, Qi = {Qis | s ∈ {1, . . , k}} is a G-transitive decomposition of [Ei ]. Moreover, for each s ∈ {1, . . , k}, GQis = GPs . Since ( , P ) is G-vertex-symmetrical, 36 for each s ∈ {1, .

A map M is said to be rotary or regular if AutM acts transitively on the darts or on the flags of M, respectively. Further, a rotary map is called chiral if it is not regular. 13. Let M be a map with underlying graph , and let G = AutM. Let P be the set of cycles which are boundaries of faces of M. Then P is a 2-uniform cycle cover of the underlying graph . If M is regular, then P is a G-arc-symmetrical cycle cover; if M is chiral, then P is a G-edge-symmetrical but not G-arc-symmetrical cover.