4-valent plane graphs with 2-, 3- and 4-gonal faces
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Call {\em i-hedrite} any 4-valent n-vertex plane graph, whose faces are 2-, 3- and 4-gons only and $p_2+p_3=i$. The edges of an i-hedrite, as of any Eulerian plane graph, are partitioned by its {\em central circuits}, i.e. those, which are obtained by starting with an edge and continuing at each vertex by the edge opposite the entering one. So, any i-hedrite is a projection of an alternating link, whose components correspond to its central circuits.
Call an i-hedrite {\em irreducible}, if it has no {\em rail-road}, i.e. a circuit of 4-gonal faces, in which every 4-gon is adjacent to two of its neighbors on opposite edges.
We present the list of all i-hedrites with at most 15 vertices. Examples of other results:
(i) All i-hedrites, which are not 3-connected, are identified. (ii) Any irreducible i-hedrite has at most i-2 central circuits. (iii) All i-hedrites without self-intersecting central circuits are listed. (iv) All symmetry group of i-hedrites are listed.
27 pages, 16 figures, submitted to satellite conference (of ISM congress, (2002) Beijing) on Algebra and Combinatorics
27 pages, 16 figures, submitted to satellite conference (of ISM congress, (2002) Beijing) on Algebra and Combinatorics