## Return to Answer

2 fixed latex

Here is another generalization of planar graphs.

Start with a d-dimensional polytope P with n vertices. For every 2-dimensional face F triangulate F by non crossing diagonals. So if F has k sides you add (k-3) edges. It is known that the total number of edges you get (including the original edges of the polytope) is at least $dn - {{d+1} \choose {2}$. 2}}\$. A polytope is called "elementary" if equality holds.

We can consider the following classes of graphs:

1) E_d = Graphs of elementary d-polytopes and all their subgraphs

2) F_d = Graphs obtained by elementary d-polytopes by triangulating all 2 faces by non crossing diagonals, and all their subgraphs.

For d=3 both classes are the class of planar graphs.

Some properties of planar graphs are known or conjectured to extend.

1) (robustness; conjectured) We can start instead of polytopes by arbitrary polyhedral (d-1)-dimensional pseudomanifolds. But it is conjectured that we will get precisely the same class of graphs.

2) (duality; known) If P is elementary so is its dual P*

3) (coloring; conjectured) Graphs in E_d (and perhaps even in F_d) are (d+1)-colorable.

1

Here is another generalization of planar graphs.

Start with a d-dimensional polytope P with n vertices. For every 2-dimensional face F triangulate F by non crossing diagonals. So if F has k sides you add (k-3) edges. It is known that the total number of edges you get (including the original edges of the polytope) is at least $dn - {{d+1} \choose {2}$. A polytope is called "elementary" if equality holds.

We can consider the following classes of graphs:

1) E_d = Graphs of elementary d-polytopes and all their subgraphs

2) F_d = Graphs obtained by elementary d-polytopes by triangulating all 2 faces by non crossing diagonals, and all their subgraphs.

For d=3 both classes are the class of planar graphs.

Some properties of planar graphs are known or conjectured to extend.

1) (robustness; conjectured) We can start instead of polytopes by arbitrary polyhedral (d-1)-dimensional pseudomanifolds. But it is conjectured that we will get precisely the same class of graphs.

2) (duality; known) If P is elementary so is its dual P*

3) (coloring; conjectured) Graphs in E_d (and perhaps even in F_d) are (d+1)-colorable.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.944796085357666, "perplexity_flag": "middle"}