Strengthening of a theorem about 3-polytopes

✍ Scribed by E. Jucovič

• Publisher: Springer
• Year: 1974
• Tongue: English
• Weight: 197 KB
• Volume: 3
• Category: Article
• ISSN: 0046-5755
• DOI: 10.1007/bf00183214

✦ Synopsis

A. Kotzig [5] proved the following theorem (cf. B. Griinbaum [2,3,4]: Every 3-polytope has at least one edge such that the sum of valencies of its end-vertices is ~< 13.

In this note we deal with improvements of this statement. Let us review first some of the notations employed: If we are given a polytope M,f(M) or v i (M) or e~, j(M) as well as simplyf or v~ or e~, j denotes the number of faces or the number of i-valent vertices or the number of edges with endvertices of valencies i, j of the polytope M, respectively. -Our main result is THEOREM. For a simplicial 3-polytope the following relation holds 120 ~< 20e3.a + 25e3,, + 16e3, 5 + 10ca, 6 + 20/3e3, 7 + 5ea, 8 + + 5/2e3,9 + 2e3,1o + 20e4,4 + 11e4,5 + 5e4,6 + 5e4,7 + + 5e4, 8 + 3e4, 9 + 8e5,5 + 2e5,6 + 2e5, 7 + 2e5, s.