- Email: [email protected]
Describing the neighborhoods of 5-vertices in 3-polytopes with minimum degree 5 and no vertices of degree from 6 to 8
Discrete Mathematics 342 (2019) 2439–2444
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Describing the neighborhoods of 5-vertices in 3-polytopes with minimum degree 5 and no vertices of degree from 6 to 8✩ O.V. Borodin, A.O. Ivanova, O.N. Kazak Sobolev Institute of Mathematics, Novosibirsk 630090, Russia
article
info
Article history: Received 23 April 2018 Received in revised form 21 April 2019 Accepted 8 May 2019 Available online 27 May 2019 Keywords: Planar graph Structural properties 3-polytope 5-star Neighborhood
a b s t r a c t In 1940, Lebesgue proved that every 3-polytope with minimum degree 5 contains a 5vertex for which the set of degrees of its neighbors is majorized by one of the following sequences: (6, 6, 7, 7, 7), (6, 6, 6, 7, 9), (6, 6, 6, 6, 11), (5, 6, 7, 7, 8), (5, 6, 6, 7, 12), (5, 6, 6, 8, 10), (5, 6, 6, 6, 17), (5, 5, 7, 7, 13), (5, 5, 7, 8, 10), (5, 5, 6, 7, 27), (5, 5, 6, 6, ∞), (5, 5, 6, 8, 15), (5, 5, 6, 9, 11), (5, 5, 5, 7, 41), (5, 5, 5, 8, 23), (5, 5, 5, 9, 17), (5, 5, 5, 10, 14), (5, 5, 5, 11, 13). Recently, we proved that forbidding vertices of degree from 7 to 11 results in a tight description (5, 5, 6, 6, ∞), (5, 6, 6, 6, 15), (6, 6, 6, 6, 6). The purpose of this note is to prove that every 3-polytope with minimum degree 5 and no vertices of degrees 6, 7, and 8 has a 5-vertex whose neighborhood is majorized by one of the sequences (5, 5, 5, 5, ∞) and (5, 5, 5, 10, 12), which is tight and improves a corresponding description (5, 5, 5, 5, ∞), (5, 5, 9, 5, 17), (5, 5, 10, 5, 14), (5, 5, 11, 5, 13) that follows from the Lebesgue Theorem. © 2019 Elsevier B.V. All rights reserved.
1. Introduction By a 3-polytope P we mean a finite 3-connected plane graph. The degree d(v ) of a vertex v (r(f ) of a face f ) in P is the number of edges incident with it. Let P5 denote the class of 3-polytopes with minimum degree 5. A k-vertex ( k-face) is a vertex (face) of degree k; a k+ -vertex has degree at least k, etc. (m) (m) By a minor k-star Sk we mean a star with k rays centered at a 5− -vertex. The weight (height) of an Sk in P is the degree sum (maximum degree) of its boundary vertices, and wk (P) (hk (P)) denotes the minimum weight (height) of minor k-stars in P. In 1904, Wernicke [24] proved that every 3-polytope in P5 has a 5-vertex adjacent to a 6− -vertex, which was strengthened by Franklin [23] in 1922 by proving that in fact there is a 5-vertex adjacent to two 6− -vertices. Recently, Borodin and Ivanova [2] proved that every 3-polytope in P5 has also a vertex of degree at most 6 adjacent to a 5-vertex and another vertex of degree at most 6, which description is tight. We say that a 5-vertex v is of type (k1 , . . . , k5 ) or simply a (k1 , . . . , k5 )-vertex if the set of degrees of its neighbors is majorized by the vector (k1 , . . . , k5 ). If the order of certain entries in the type is irrelevant, then we put a line over them. ✩ The work was funded by the Russian Science Foundation (grant 16-11-10054-P). E-mail addresses: [email protected] (O.V. Borodin), [email protected] (A.O. Ivanova), [email protected] (O.N. Kazak). https://doi.org/10.1016/j.disc.2019.05.010 0012-365X/© 2019 Elsevier B.V. All rights reserved.
2440
O.V. Borodin, A.O. Ivanova and O.N. Kazak / Discrete Mathematics 342 (2019) 2439–2444
In 1940, the following description of the neighborhoods of 5-vertices in P5 was given by Lebesgue [21, p. 36], which absorbs the results of Wernicke [24] and Franklin [23]: Theorem 1 (Lebesgue [21]). Every triangulated 3-polytope with minimum degree 5 contains a 5-vertex of one of the following types: (6, 6, 7, 7, 7), (6, 6, 6, 7, 9), (6, 6, 6, 6, 11), (5, 6, 7, 7, 8), (5, 6, 6, 7, 11), (5, 6, 6, 8, 8), (5, 6, 6, 9, 7), (5, 7, 6, 6, 12), (5, 8, 6, 6, 10), (5, 6, 6, 6, 17), (5, 5, 7, 7, 8), (5, 13, 5, 7, 7), (5, 10, 5, 7, 8), (5, 8, 5, 7, 9), (5, 7, 5, 7, 10), (5, 7, 5, 8, 8), (5, 5, 7, 6, 12), (5, 5, 8, 6, 10), (5, 6, 5, 7, 12), (5, 6, 5, 8, 10), (5, 17, 5, 6, 7), (5, 11, 5, 6, 8), (5, 11, 5, 6, 9), (5, 7, 5, 6, 13), (5, 8, 5, 6, 11), (5, 9, 5, 6, 10), (5, 6, 6, 5, ∞), (5, 5, 7, 5, 41), (5, 5, 8, 5, 23), (5, 5, 9, 5, 17), (5, 5, 10, 5, 14), (5, 5, 11, 5, 13). (m)
In particular, Theorem 1 implies that there is a 5-vertex with three 7− -neighbors, which means that h(S3 ) ≤ 7. (m) Another corollary of Theorem 1 is that w (S3 ) ≤ 24, which was improved in 1996 by Jendrol’ and Madaras [18] to the (m) sharp bound w (S3 ) ≤ 23. Furthermore, Jendrol’ and Madaras [18] gave a tight description of minor 3-stars in P5 : there (m) is a (6, 6, 6)- or (5, 6, 7)-star. Recently, Borodin and Ivanova [1], using the sharp bound w (S4 ) ≤ 30 by Borodin and Woodall [12], obtained a tight description of minor 4-stars in P5 . Jendrol’ and Madaras [18] also showed that if a polytope P in P5 is allowed to have a 5-vertex adjacent to four 5-vertices (such a 5-vertex is also called a minor (5, 5, 5, 5, ∞)-star), then h5 (P) (and hence w5 (P)) can be arbitrarily large. In 2014, Borodin, Ivanova, and Jensen [7] showed that the same phenomenon holds under a weaker assumption that 5-vertices are allowed to have two 5-neighbors and two 6-neighbors. Thus, the term (5, 6, 6, 5, ∞) in Theorem 1 is necessary. Some recent sharp bounds on the height and weight of minor 5-stars in various subclasses of P5 , along with several related results, can be found in [1–14,17,19] and surveys [5,20]. (m) In particular, Borodin, Ivanova and Nikiforov [11] obtained a sharp bound h(S5 ) ≤ 17 under the absence of 6-vertices, which improves the upper bound 41 that follows from Theorem 1. In 2013, Ivanova and Nikiforov [15] corrected two misprints in the statement of Theorem 1: 11 in (5, 11, 5, 6, 8) should be replaced by 15, and in (5, 17, 5, 6, 7) there should be 27 instead of 17. Later on, they improved [16,22] thus corrected version of Theorem 1 by replacing 41 and 23 in the types (5, 5, 7, 5, 41) and (5, 5, 8, 5, 23) to 31 and 22, respectively. Theorem 2 (Ivanova, Nikiforov [15,16,22]). Every 3-polytope with minimum degree 5 contains a 5-vertex of one of the following types: (6, 6, 7, 7, 7), (6, 6, 6, 7, 9), (6, 6, 6, 6, 11), (5, 8, 6, 7, 7), (5, 7, 6, 8, 7), (5, 6, 6, 7, 11), (5, 6, 6, 8, 8), (5, 7, 6, 6, 12), (5, 8, 6, 6, 10), (5, 6, 6, 6, 17), (5, 5, 7, 7, 8), (5, 13, 5, 7, 7), (5, 10, 5, 7, 8), (5, 8, 5, 7, 9), (5, 7, 5, 7, 10), (5, 7, 5, 8, 8), (5, 5, 7, 6, 12), (5, 5, 8, 6, 10), (5, 6, 5, 7, 12), (5, 6, 5, 8, 10), (5, 27, 5, 6, 7), (5, 15, 5, 6, 8), (5, 11, 5, 6, 9), (5, 7, 5, 6, 13), (5, 8, 5, 6, 11), (5, 9, 5, 6, 10), (5, 6, 6, 5, ∞), (5, 5, 7, 5, 31), (5, 5, 8, 5, 22), (5, 5, 9, 5, 17), (5, 5, 10, 5, 14), (5, 5, 11, 5, 13). Recently, we proved in [8] that forbidding vertices of degree from 7 to 11 in P5 results in a tight description (5, 5, 6, 6, ∞), (5, 6, 6, 6, 15), (6, 6, 6, 6, 6), which improves a description (5, 5, 6, 6, ∞), (5, 6, 6, 6, 17), (6, 6, 6, 6, 6) that follows from Theorem 1. If vertices of degrees 6, 7, and 8 are forbidden, then Theorem 1 implies a 5-vertex of one of the following types: (5, 5, 5, 5, ∞), (5, 5, 9, 5, 17), (5, 5, 10, 5, 14), (5, 5, 11, 5, 13). The purpose of this note is to obtain a precise description of 5-stars in this subclass of P5 . Theorem 3. Every 3-polytope with minimum degree 5 and without vertices of degree from 6 to 8 has a 5-vertex of one of the types (5, 5, 5, 5, ∞) and (5, 5, 10, 5, 12), where all parameters are best possible. 2. Proving Theorem 3 To confirm the tightness of the term (5, 5, 10, 5, 12), we start with the (5, 6, 6)-Archimedean solid, which is a cubic 3-polytope whose each vertex is incident with a 5-face and two 6-faces, replace all its vertices by small 3-faces, and cap each 10+ -face obtained.
O.V. Borodin, A.O. Ivanova and O.N. Kazak / Discrete Mathematics 342 (2019) 2439–2444
2441
The resulting 3-polytope has only 5-vertices, 10-vertices, and 12-vertices, and all 5-vertices are of type (5, 5, 10, 5, 12) or (5, 5, 12, 5, 12), as desired. The construction confirming the tightness of (5, 5, 5, 5, ∞) is due to Jendrol’ and Madaras [18]. 2.1. Discharging Suppose that a 3-polytope P5′ is a counterexample to the main statement of Theorem 3. Thus each 5-vertex in P5′ is adjacent either to at most two 5-vertices, or otherwise to precisely two consecutive 9+ -vertices, or a 9-vertex non-consecutive with a 13+ -vertex, or a 10-vertex non-consecutive with a 13+ -vertex, or two non-consecutive 11+ -vertices. Let P5 be a counterexample with the maximum number of edges having the same number of vertices as P5′ . Remark 1. P5 has no 4+ -face with two non-consecutive 9+ -vertices along the boundary, for otherwise adding a diagonal between these vertices would result in a counterexample with more edges than P5 . Let V , E, and F be the sets of vertices, edges, and faces of P5 . Euler’s formula |V | − |E | + |F | = 2 implies
∑
∑
v∈V
f ∈F
(d(v ) − 6) +
(2d(f ) − 6) = −12.
(1)
We assign an initial charge µ(v ) = d(v ) − 6 to each v ∈ V and µ(f ) = 2d(f ) − 6 to each f ∈ F , so that only 5-vertices have negative initial charge. Using the properties of P5 as a counterexample to Theorem 3, we define a local redistribution of charges, preserving their sum, such that the final charge µ(x) is non-negative for all x ∈ V ∪ F . This will contradict the fact that the sum of the final charges is, by (1), equal to −12. The final charge µ′ (x) whenever x ∈ V ∪ F is defined by applying the rules R1–R7 below (see Fig. 1). For a vertex v , let v1 , . . . , vd(v ) be the vertices adjacent to v in a fixed cyclic order. A vertex is simplicial if it is completely surrounded by 3-faces. A 5-vertex v is strong if d(v1 ) = d(v2 ) = 5, d(v3 ) ≥ 9, d(v4 ) ≥ 9, d(v5 ) ≥ 9, and there is a 3-face vv1 v2 . Note that v also is incident to 3-faces v3 vv4 and v4 vv5 due to Remark 1. A simplicial 5-vertex v such that d(v1 ) = d(v2 ) = d(v4 ) = 5, 9 ≤ d(v3 ) ≤ 10, and d(v5 ) ≥ 13 is poor, and v1 is paired with v . We note that the poor and paired neighbors in the neighborhood of each 13+ -vertex w are in one-to-one correspondence with each other. Indeed, if w2 were paired with two poor vertices w1 and w3 , then w2 would have four 5-neighbors, a contradiction. On the other hand, if w1 , w2 , w3 are poor neighbors of w , where w1 and w2 have a common neighbor of degree 9 or 10, then w2 is paired with w3 , but not with w1 due to a unique 3-face incident with three 5-vertices at a poor vertex. R1. A 4+ -face f = v1 . . . vd(f ) gives each incident 5-vertex v2 : (a) 21 if d(v1 ) = d(v3 ) = 5, or (b) 34 if d(v1 ) ≥ 9 and d(v3 ) = 5. R2. A 5-vertex v with d(v1 ) ≥ 9 receives the following charge from its 9+ -neighbor v2 : 7 (a) if d(v3 ) = 5, then 12 in the case that d(v2 ) ≤ 12 and 12 otherwise, or 1 (b) 2 if d(v3 ) ≥ 9. R3. A non-simplicial 5-vertex v with d(v1 ) = d(v3 ) = d(v4 ) = 5 receives 41 from each of its 9+ -neighbors v2 and v5 . R4. A strong 5-vertex gives 16 to each of its two 5-neighbors. R5. A simplicial 5-vertex v with d(v1 ) = d(v2 ) = d(v4 ) = 5 receives from v5 : (a) 31 if 9 ≤ d(v5 ) ≤ 10, and (b) 12 if 11 ≤ d(v5 ) ≤ 12. R6. If a simplicial vertex v satisfies d(v1 ) = d(v2 ) = d(v4 ) = 5 and d(v5 ) ≥ 13, then v5 gives 21 to v , with the following two exceptions: 7 (ex1) if 9 ≤ d(v3 ) ≤ 10 and v2 has three 5-neighbors, then v5 gives 12 to v ; 7 (ex2) if d(v3 ) ≥ 13, v is paired with a poor vertex v1 , and v2 has three 5-neighbors, then v5 gives 12 to v . 1 R7. A poor 5-vertex v receives 12 from its paired vertex v1 if v2 has three 5-neighbors. 2.2. Checking µ′ (x) ≥ 0 whenever x ∈ V ∪ F Note that a 3-face f does not participate in discharging, so µ′ (f ) = µ(f ) = 2 × 3 − 6 = 0. Now if f is a 4+ -face, then its donation of 34 by R1b may be interpreted as giving 12 to the 5-vertex and 41 to the neighbor 9+ -vertex along the boundary 3(d(f )−4) of f . As a result, each vertex incident with f receives at most 12 from f , so µ′ (f ) ≥ 2d(f ) − 6 − d(f ) × 21 = ≥ 0. 2 Now suppose v ∈ V . Case 1. d(v ) = 5. If v is adjacent to at least four 9+ -vertices, then µ′ (v ) ≥ 5 − 6 + 4 × 21 > 0 by R2. Suppose v has precisely three 9+ -neighbors. If they are consecutive, then v receives at least 3 × 21 from them by 1 R2 and can give 2 × 16 to the two 5-neighbors by R4 if v is strong, and 12 to the paired 5-vertex by R7, which implies
2442
O.V. Borodin, A.O. Ivanova and O.N. Kazak / Discrete Mathematics 342 (2019) 2439–2444
Fig. 1. Rules of discharging.
1 µ′ (v ) ≥ 3 × 21 − 2 × 61 − 12 > 0 in view of the one-to-one correspondence between poor and paired vertices. Now if d(v1 ) = d(v3 ) = 5, then v receives at least 12 from each of v4 and v5 by R2a. Since v is not strong, it gives no charge to v1 and v3 by R4. Also v cannot give charge by R7, since it has no consecutive 5-neighbors as we see in R7. So, we have µ′ (v ) ≥ −1 + 2 × 12 = 0. Since v cannot have four 5-neighbors, it remains to assume that v has precisely two 9+ -neighbors. First suppose d(v1 ) ≥ d(v2 ) ≥ 9. If d(v1 ) ≤ 12, then v receives 12 from each of v1 and v2 by R2a and gives nothing away by R7, so 1 7 to v by R2a. In turn, v gives to v5 either 12 by R7 if v is paired with v5 or no charge µ′ = 0. If d(v1 ) ≥ 13, then v1 gives 12 7 1 otherwise. In both cases, v1 brings at least 21 = 12 − 12 to v in total. The 9+ -vertex v2 also brings at least 12 to v by R2a possibly combined with R7, and hence µ′ (v ) ≥ −1 + 2 × 21 = 0. From now on we can assume that d(v1 ) ≥ 11 and d(v3 ) ≥ 9 due to the absence of (5, 5, 10, 5, 10)-vertices. If v is not simplicial, then v receives 2 × 41 from v1 and v3 by R3 and at least 21 from an incident 4+ -face by R1. We already have 1 µ′ (v ) ≥ 0 unless v gives 12 to at least one of v4 and v5 by R7. This happens only if the face f = · · · v4 vv5 is a triangle, in 1 which case v actually receives 43 by R1b from an incident 4+ -face, so µ′ (v ) ≥ −1 + 34 + 2 × 14 − 2 × 12 > 0. 1 Finally, suppose v is simplicial. If v gives 12 to v5 by R7, so that v is paired with a poor vertex v5 , then d(v1 ) ≥ 13 and v4 is not strong. The latter implies that v4 has a 5-neighbor different from v and v5 . In turn, this means that d(v3 ) ≥ 13 since otherwise v would be a (5, 5, 10, 5, 12)-vertex, a contradiction. 7 7 1 Therefore, v receives 12 from v1 by R6ex2, and hence v1 brings 12 − 12 to v . We note that v3 also brings 12 to v either 1 ′ by R6 alone or by R6ex2 combined with R7. Thus we have µ (v ) = −1 + 2 × 12 = 0 when v gives away 12 at least once
in R7.
O.V. Borodin, A.O. Ivanova and O.N. Kazak / Discrete Mathematics 342 (2019) 2439–2444
2443
1 From now we can assume that v is not a donator of 12 by R7. Each 11+ -neighbor gives v at least 12 by R5b and R6, so it remains to assume that d(v3 ) ≤ 10. This implies that d(v1 ) ≥ 13 due to the absence of (5, 5, 10, 5, 12)-vertices by assumption. 1 from v5 by R7 too), 13 from v3 by Thus our v is poor. If v5 is strong, then v receives 61 from v5 by R4 (and maybe 12 1 1 7 ′ R5a, and 2 from v1 by R6, which results in µ (v ) ≥ 0. Otherwise, v receives 12 from v5 by R7, 13 from v3 , and 12 from v1 by R6ex1, so µ′ (v ) = 0. Case 2. 9 ≤ d(v ) ≤ 10. To estimate the total donation of v by R2, R3 and R5a, we average donations of v to its 5-neighbors according to R2a,b as follows. Instead of giving 12 to a 5-neighbor v2 by R2b, let v give 61 to each of the 9+ -vertices v1 and v3 , so the modified donation to v2 becomes 61 . If d(v1 ) = d(v2 ) = 5 and d(v3 ) ≥ 9, as in R2a, then v instead can give 13 to v2 and 16 to the 9+ -vertex v3 . As a result, each neighbor receives at most 13 = 16 + 16 = 21 − 16 from v , so we see that µ′ (v ) ≥ d(v ) − 6 − d(3v) = 2(d(v3)−9) ≥ 0. d(v ) Case 3. 11 ≤ d(v ) ≤ 12. We note that v gives each neighbor at most 21 by R2, R3 and R5b, so µ′ (v ) ≥ d(v ) − 6 − 2 = d(v )−12 , which settles the case d(v ) = 12. 2 So suppose d(v ) = 11. If v has a 9+ -neighbor, then µ′ (v ) ≥ 11 − 6 − 10 × 12 = 0. Thus we can assume that v is completely surrounded by 5-vertices. If v is incident with a 4+ -face . . . v1 vv2 , then each of v1 and v2 receives nothing by R2 and at most 41 by R3 from v . This implies µ′ (v ) ≥ 5 − 2 × 14 − (11 − 2) × 21 = 0 in view of R5b. It remains to assume in addition that v is simplicial. Now if there is a 4+ -face . . . v1′ v1 v2 v2′ , then still each of v1 and v2 receives at most 41 from v by R3, so µ′ (v ) ≥ 0 as above. Thus we are done unless there are vertices w1 , . . . , w11 such that there are 3-faces wk vk vk+1 whenever 1 ≤ k ≤ 11 (addition modulo 11). If so, then we cannot have d(wk ) = d(wk+1 ) = 5 for any k due to the absence of (5, 5, 5, 5, ∞)vertices. By the oddness of 11, this implies that, say, d(w1 ) ≥ 9 and d(w2 ) ≥ 9, in which case v gives no charge to v2 by R2, R3 and R5b, which yields µ′ (v ) ≥ 5 − 10 × 12 = 0. 7 Case 4. d(v ) ≥ 13. We know that v gives at most 12 to each adjacent 5-vertex by R2, R3, and R6. Since µ(v ) = 7d(v ) 5d(v )−72 2 7 ′ d(v ) − 6 − 12 = , it follows that µ ( v ) > 0 for d(v ) ≥ 15, µ′ (v ) ≥ − 12 for d(v ) = 14, and µ′ (v ) ≥ − 12 12 for d(v ) = 13. Thus we are already done for d(v ) ≥ 15 and now we wish to improve these rough estimations for 13 ≤ d(v ) ≤ 14. 7(d(v )−1) We have nothing to prove unless v is completely surrounded by 5-vertices, for otherwise µ′ (v ) ≥ d(v ) − 6 − = 12 7(d(v )−2) 5(d(v )−13) 1 ′ ≥ 0. Also, the argument used in Case 3 shows that if v is not simplicial, then µ ( v ) ≥ d( v ) − 6 − 2 × − = 12 4 12 5(d(v )−13) 1 1 ′ ′ + ′ + > 0. Now if there is a 4 -face . . . v v v v , then we similarly have µ ( v ) ≥ due to R3. 1 2 1 2 12 12 12 As in Case 3, we now have a cyclic sequence Wd(v ) = w1 , . . . , wd(v ) such that there are 3-faces wk vk vk+1 (addition modulo d(v )) whenever 1 ≤ k ≤ d(v ). Again, there are no two consecutive 5-vertices in Wd(v ) since each vk must have a 9+ -neighbor other than v . If there are two consecutive 9+ -vertices in Wd(v ) , say w1 and w2 , then v2 receives no charge from v by R1–R7, so we 7 7 7 can improve our rough estimation µ′ (v ) ≥ − 12 to µ′ (v ) ≥ − 12 + 12 ≥ 0, as desired. Due to the oddness of 13, this completes the proof for d(v ) = 13. So suppose d(v ) = 14. If v has at least one non-simplicial 5-neighbor vk , then vk receives at most 14 from v by R3, so 7 ′ µ (v ) ≥ 8 − 14 − 13 × 12 > 0. Thus we can assume that v is completely surrounded by simplicial 5-vertices. This means that W14 is actually a 14-cycle, and we can assume that d(w1 ) = d(w3 ) = · · · = d(w13 ) = 5. Now if at least one of 5-vertices in W14 , say w1 , is strong, that is w1 has a 9+ -neighbor outside W14 , then each of v1 7 7 and v2 receives 12 by R6 rather than 12 by R6ex1 or R6ex2, which yields µ′ (v ) ≥ 8 − 2 × 12 − 12 × 12 = 0. Now all w1 , w3 , . . . , w13 are non-strong, that is each of them has three 5-neighbors. Among the seven 9+ -vertices w2 , w4 , . . . , w14 there are no two consecutive (cyclically) 10− -vertices, for otherwise we would have a (5, 5, 10, 5, 10)vertex, which is impossible. By parity reasons and symmetry, we can assume that d(w14 ) ≥ 11 and d(w2 ) ≥ 11. So each of v1 and v2 obeys the 7 general rule R6 rather than its exceptions R6ex1 or R6ex2. This means that again µ′ (v ) ≥ 8 − 2 × 21 − 12 × 12 = 0, as desired. Thus we have proved µ′ (x) ≥ 0 whenever x ∈ V ∪ F , which contradicts (1) and completes the proof of Theorem 3.
References [1] O.V. Borodin, A.O. Ivanova, Describing 4-stars at 5-vertices in normal plane maps with minimum degree 5, Discrete Math. 313 (17) (2013) 1710–1714. [2] O.V. Borodin, A.O. Ivanova, An analogue of franklin’s theorem, Discrete Math. 339 (10) (2016) 2553–2556. [3] O.V. Borodin, A.O. Ivanova, Light and low 5-stars in normal plane maps with minimum degree 5, Siberian Math. J. 57 (3) (2016) 470–475. [4] O.V. Borodin, A.O. Ivanova, Low 5-stars in normal plane maps with minimum degree 5, Discrete Math. 340 (2) (2017) 18–22. [5] O.V. Borodin, A.O. Ivanova, New results about the structure of plane graphs: a survey, AIP Conf. Proc. 1907 (2017) 030051. [6] O.V. Borodin, A.O. Ivanova, On light neighborhoods of 5-vertices in 3-polytopes with minimum degree 5, Discrete Math. 340 (9) (2017) 2234–2242. [7] O.V. Borodin, A.O. Ivanova, T.R. Jensen, 5-stars of low weight in normal plane maps with minimum degree 5, Discuss. Math. Graph Theory 34 (3) (2014) 539–546. [8] O.V. Borodin, A.O. Ivanova, O.N. Kazak, Describing neighborhoods of 5-vertices in 3-polytopes with minimum degree 5 and without vertices of degrees from 7 to 11, Discuss. Math. Graph Theory 38 (3) (2018) 615–625.
2444
O.V. Borodin, A.O. Ivanova and O.N. Kazak / Discrete Mathematics 342 (2019) 2439–2444
[9] O.V. Borodin, A.O. Ivanova, O.N. Kazak, E.I. Vasil’eva, Heights of minor 5-stars in 3-polytopes with minimum degree 5 and no vertices of degree 6 and 7, Discrete Math. 341 (3) (2018) 825–829. [10] O.V. Borodin, A.O. Ivanova, D.V. Nikiforov, Low and light 5-stars in 3-polytopes with minimum degree 5 and restrictions on the degrees of major vertices, Siberian Math. J. 58 (4) (2017) 600–605. [11] O.V. Borodin, A.O. Ivanova, D.V. Nikiforov, Low minor 5-stars in 3-polytopes with minimum degree 5 and no 6-vertices, Discrete Math. 340 (7) (2017) 1612–1616. [12] O.V. Borodin, D.R. Woodall, Short cycles of low weight in normal plane maps with minimum degree 5, Discuss. Math. Graph Theory 18 (2) (1998) 159–164. [13] B. Ferencová, T. Madaras, Light graph in families of polyhedral graphs with prescribed minimum degree, face size, edge and dual edge weight, Discrete Math. 310 (12) (2010) 1661–1675. [14] J. Harant, S. Jendrol’, On the existence of specific stars in planar graphs, Graphs Combin. 23 (2007) 529–543. [15] A.O. Ivanova, D.V. Nikiforov, The structure of neighborhoods of 5-vertices in plane triangulation with minimum degree 5 (Russian), Math. Notes Yakutsk State Univ. 20 (2) (2013) 66–78. [16] A.O. Ivanova, D.V. Nikiforov, Combinatorial structure of triangulated 3-polytopes with minimum degree 5 (Russian), in: Proceedings of the Scientific Conference of Students, Graduate Students, and Young Researchers. XVII and XVIII Lavrent’ev’s Reading, International Center for Research Projects, Yakutsk; Kirov, 2015, pp. 22–27. [17] S. Jendrol’, A structural property of convex 3-polytopes, Geom. Dedicata 68 (1997) 91–99. [18] S. Jendrol’, T. Madaras, On light subgraphs in plane graphs of minimal degree five, Discuss. Math. Graph Theory 16 (1996) 207–217. [19] S. Jendrol’, T. Madaras, Note on an existence of small degree vertices with at most one big degree neighbour in planar graphs, Tatra Mt. Math. Publ. 30 (2005) 149–153. [20] S. Jendrol’, H.-J. Voss, Light subgraphs of graphs embedded in the plane — a survey, Discrete Math. 313 (4) (2013) 406–421. [21] H. Lebesgue, Quelques conséquences simples de la formule d’Euler, J. Math. Pures Appl. 19 (1940) 27–43. [22] D.V. Nikiforov, The structure of neighborhoods of normal plane maps with minimum degree 5 (Russian), Math. Notes North-Eastern Federal Univ. 23 (1) (2016) 56–66. [23] Ph. Franklin, The four colour problem, Amer. J. Math. 44 (1922) 225–236. [24] P. Wernicke, Über den kartographischen Vierfarbensatz, Math. Ann. 58 (1904) 413–426.
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Describing the neighborhoods of 5-vertices in 3-polytopes with minimum degree 5 and no vertices of degree from 6 to 8✩ O.V. Borodin, A.O. Ivanova, O.N. Kazak Sobolev Institute of Mathematics, Novosibirsk 630090, Russia
article
info
Article history: Received 23 April 2018 Received in revised form 21 April 2019 Accepted 8 May 2019 Available online 27 May 2019 Keywords: Planar graph Structural properties 3-polytope 5-star Neighborhood
a b s t r a c t In 1940, Lebesgue proved that every 3-polytope with minimum degree 5 contains a 5vertex for which the set of degrees of its neighbors is majorized by one of the following sequences: (6, 6, 7, 7, 7), (6, 6, 6, 7, 9), (6, 6, 6, 6, 11), (5, 6, 7, 7, 8), (5, 6, 6, 7, 12), (5, 6, 6, 8, 10), (5, 6, 6, 6, 17), (5, 5, 7, 7, 13), (5, 5, 7, 8, 10), (5, 5, 6, 7, 27), (5, 5, 6, 6, ∞), (5, 5, 6, 8, 15), (5, 5, 6, 9, 11), (5, 5, 5, 7, 41), (5, 5, 5, 8, 23), (5, 5, 5, 9, 17), (5, 5, 5, 10, 14), (5, 5, 5, 11, 13). Recently, we proved that forbidding vertices of degree from 7 to 11 results in a tight description (5, 5, 6, 6, ∞), (5, 6, 6, 6, 15), (6, 6, 6, 6, 6). The purpose of this note is to prove that every 3-polytope with minimum degree 5 and no vertices of degrees 6, 7, and 8 has a 5-vertex whose neighborhood is majorized by one of the sequences (5, 5, 5, 5, ∞) and (5, 5, 5, 10, 12), which is tight and improves a corresponding description (5, 5, 5, 5, ∞), (5, 5, 9, 5, 17), (5, 5, 10, 5, 14), (5, 5, 11, 5, 13) that follows from the Lebesgue Theorem. © 2019 Elsevier B.V. All rights reserved.
1. Introduction By a 3-polytope P we mean a finite 3-connected plane graph. The degree d(v ) of a vertex v (r(f ) of a face f ) in P is the number of edges incident with it. Let P5 denote the class of 3-polytopes with minimum degree 5. A k-vertex ( k-face) is a vertex (face) of degree k; a k+ -vertex has degree at least k, etc. (m) (m) By a minor k-star Sk we mean a star with k rays centered at a 5− -vertex. The weight (height) of an Sk in P is the degree sum (maximum degree) of its boundary vertices, and wk (P) (hk (P)) denotes the minimum weight (height) of minor k-stars in P. In 1904, Wernicke [24] proved that every 3-polytope in P5 has a 5-vertex adjacent to a 6− -vertex, which was strengthened by Franklin [23] in 1922 by proving that in fact there is a 5-vertex adjacent to two 6− -vertices. Recently, Borodin and Ivanova [2] proved that every 3-polytope in P5 has also a vertex of degree at most 6 adjacent to a 5-vertex and another vertex of degree at most 6, which description is tight. We say that a 5-vertex v is of type (k1 , . . . , k5 ) or simply a (k1 , . . . , k5 )-vertex if the set of degrees of its neighbors is majorized by the vector (k1 , . . . , k5 ). If the order of certain entries in the type is irrelevant, then we put a line over them. ✩ The work was funded by the Russian Science Foundation (grant 16-11-10054-P). E-mail addresses: [email protected] (O.V. Borodin), [email protected] (A.O. Ivanova), [email protected] (O.N. Kazak). https://doi.org/10.1016/j.disc.2019.05.010 0012-365X/© 2019 Elsevier B.V. All rights reserved.
2440
O.V. Borodin, A.O. Ivanova and O.N. Kazak / Discrete Mathematics 342 (2019) 2439–2444
In 1940, the following description of the neighborhoods of 5-vertices in P5 was given by Lebesgue [21, p. 36], which absorbs the results of Wernicke [24] and Franklin [23]: Theorem 1 (Lebesgue [21]). Every triangulated 3-polytope with minimum degree 5 contains a 5-vertex of one of the following types: (6, 6, 7, 7, 7), (6, 6, 6, 7, 9), (6, 6, 6, 6, 11), (5, 6, 7, 7, 8), (5, 6, 6, 7, 11), (5, 6, 6, 8, 8), (5, 6, 6, 9, 7), (5, 7, 6, 6, 12), (5, 8, 6, 6, 10), (5, 6, 6, 6, 17), (5, 5, 7, 7, 8), (5, 13, 5, 7, 7), (5, 10, 5, 7, 8), (5, 8, 5, 7, 9), (5, 7, 5, 7, 10), (5, 7, 5, 8, 8), (5, 5, 7, 6, 12), (5, 5, 8, 6, 10), (5, 6, 5, 7, 12), (5, 6, 5, 8, 10), (5, 17, 5, 6, 7), (5, 11, 5, 6, 8), (5, 11, 5, 6, 9), (5, 7, 5, 6, 13), (5, 8, 5, 6, 11), (5, 9, 5, 6, 10), (5, 6, 6, 5, ∞), (5, 5, 7, 5, 41), (5, 5, 8, 5, 23), (5, 5, 9, 5, 17), (5, 5, 10, 5, 14), (5, 5, 11, 5, 13). (m)
In particular, Theorem 1 implies that there is a 5-vertex with three 7− -neighbors, which means that h(S3 ) ≤ 7. (m) Another corollary of Theorem 1 is that w (S3 ) ≤ 24, which was improved in 1996 by Jendrol’ and Madaras [18] to the (m) sharp bound w (S3 ) ≤ 23. Furthermore, Jendrol’ and Madaras [18] gave a tight description of minor 3-stars in P5 : there (m) is a (6, 6, 6)- or (5, 6, 7)-star. Recently, Borodin and Ivanova [1], using the sharp bound w (S4 ) ≤ 30 by Borodin and Woodall [12], obtained a tight description of minor 4-stars in P5 . Jendrol’ and Madaras [18] also showed that if a polytope P in P5 is allowed to have a 5-vertex adjacent to four 5-vertices (such a 5-vertex is also called a minor (5, 5, 5, 5, ∞)-star), then h5 (P) (and hence w5 (P)) can be arbitrarily large. In 2014, Borodin, Ivanova, and Jensen [7] showed that the same phenomenon holds under a weaker assumption that 5-vertices are allowed to have two 5-neighbors and two 6-neighbors. Thus, the term (5, 6, 6, 5, ∞) in Theorem 1 is necessary. Some recent sharp bounds on the height and weight of minor 5-stars in various subclasses of P5 , along with several related results, can be found in [1–14,17,19] and surveys [5,20]. (m) In particular, Borodin, Ivanova and Nikiforov [11] obtained a sharp bound h(S5 ) ≤ 17 under the absence of 6-vertices, which improves the upper bound 41 that follows from Theorem 1. In 2013, Ivanova and Nikiforov [15] corrected two misprints in the statement of Theorem 1: 11 in (5, 11, 5, 6, 8) should be replaced by 15, and in (5, 17, 5, 6, 7) there should be 27 instead of 17. Later on, they improved [16,22] thus corrected version of Theorem 1 by replacing 41 and 23 in the types (5, 5, 7, 5, 41) and (5, 5, 8, 5, 23) to 31 and 22, respectively. Theorem 2 (Ivanova, Nikiforov [15,16,22]). Every 3-polytope with minimum degree 5 contains a 5-vertex of one of the following types: (6, 6, 7, 7, 7), (6, 6, 6, 7, 9), (6, 6, 6, 6, 11), (5, 8, 6, 7, 7), (5, 7, 6, 8, 7), (5, 6, 6, 7, 11), (5, 6, 6, 8, 8), (5, 7, 6, 6, 12), (5, 8, 6, 6, 10), (5, 6, 6, 6, 17), (5, 5, 7, 7, 8), (5, 13, 5, 7, 7), (5, 10, 5, 7, 8), (5, 8, 5, 7, 9), (5, 7, 5, 7, 10), (5, 7, 5, 8, 8), (5, 5, 7, 6, 12), (5, 5, 8, 6, 10), (5, 6, 5, 7, 12), (5, 6, 5, 8, 10), (5, 27, 5, 6, 7), (5, 15, 5, 6, 8), (5, 11, 5, 6, 9), (5, 7, 5, 6, 13), (5, 8, 5, 6, 11), (5, 9, 5, 6, 10), (5, 6, 6, 5, ∞), (5, 5, 7, 5, 31), (5, 5, 8, 5, 22), (5, 5, 9, 5, 17), (5, 5, 10, 5, 14), (5, 5, 11, 5, 13). Recently, we proved in [8] that forbidding vertices of degree from 7 to 11 in P5 results in a tight description (5, 5, 6, 6, ∞), (5, 6, 6, 6, 15), (6, 6, 6, 6, 6), which improves a description (5, 5, 6, 6, ∞), (5, 6, 6, 6, 17), (6, 6, 6, 6, 6) that follows from Theorem 1. If vertices of degrees 6, 7, and 8 are forbidden, then Theorem 1 implies a 5-vertex of one of the following types: (5, 5, 5, 5, ∞), (5, 5, 9, 5, 17), (5, 5, 10, 5, 14), (5, 5, 11, 5, 13). The purpose of this note is to obtain a precise description of 5-stars in this subclass of P5 . Theorem 3. Every 3-polytope with minimum degree 5 and without vertices of degree from 6 to 8 has a 5-vertex of one of the types (5, 5, 5, 5, ∞) and (5, 5, 10, 5, 12), where all parameters are best possible. 2. Proving Theorem 3 To confirm the tightness of the term (5, 5, 10, 5, 12), we start with the (5, 6, 6)-Archimedean solid, which is a cubic 3-polytope whose each vertex is incident with a 5-face and two 6-faces, replace all its vertices by small 3-faces, and cap each 10+ -face obtained.
O.V. Borodin, A.O. Ivanova and O.N. Kazak / Discrete Mathematics 342 (2019) 2439–2444
2441
The resulting 3-polytope has only 5-vertices, 10-vertices, and 12-vertices, and all 5-vertices are of type (5, 5, 10, 5, 12) or (5, 5, 12, 5, 12), as desired. The construction confirming the tightness of (5, 5, 5, 5, ∞) is due to Jendrol’ and Madaras [18]. 2.1. Discharging Suppose that a 3-polytope P5′ is a counterexample to the main statement of Theorem 3. Thus each 5-vertex in P5′ is adjacent either to at most two 5-vertices, or otherwise to precisely two consecutive 9+ -vertices, or a 9-vertex non-consecutive with a 13+ -vertex, or a 10-vertex non-consecutive with a 13+ -vertex, or two non-consecutive 11+ -vertices. Let P5 be a counterexample with the maximum number of edges having the same number of vertices as P5′ . Remark 1. P5 has no 4+ -face with two non-consecutive 9+ -vertices along the boundary, for otherwise adding a diagonal between these vertices would result in a counterexample with more edges than P5 . Let V , E, and F be the sets of vertices, edges, and faces of P5 . Euler’s formula |V | − |E | + |F | = 2 implies
∑
∑
v∈V
f ∈F
(d(v ) − 6) +
(2d(f ) − 6) = −12.
(1)
We assign an initial charge µ(v ) = d(v ) − 6 to each v ∈ V and µ(f ) = 2d(f ) − 6 to each f ∈ F , so that only 5-vertices have negative initial charge. Using the properties of P5 as a counterexample to Theorem 3, we define a local redistribution of charges, preserving their sum, such that the final charge µ(x) is non-negative for all x ∈ V ∪ F . This will contradict the fact that the sum of the final charges is, by (1), equal to −12. The final charge µ′ (x) whenever x ∈ V ∪ F is defined by applying the rules R1–R7 below (see Fig. 1). For a vertex v , let v1 , . . . , vd(v ) be the vertices adjacent to v in a fixed cyclic order. A vertex is simplicial if it is completely surrounded by 3-faces. A 5-vertex v is strong if d(v1 ) = d(v2 ) = 5, d(v3 ) ≥ 9, d(v4 ) ≥ 9, d(v5 ) ≥ 9, and there is a 3-face vv1 v2 . Note that v also is incident to 3-faces v3 vv4 and v4 vv5 due to Remark 1. A simplicial 5-vertex v such that d(v1 ) = d(v2 ) = d(v4 ) = 5, 9 ≤ d(v3 ) ≤ 10, and d(v5 ) ≥ 13 is poor, and v1 is paired with v . We note that the poor and paired neighbors in the neighborhood of each 13+ -vertex w are in one-to-one correspondence with each other. Indeed, if w2 were paired with two poor vertices w1 and w3 , then w2 would have four 5-neighbors, a contradiction. On the other hand, if w1 , w2 , w3 are poor neighbors of w , where w1 and w2 have a common neighbor of degree 9 or 10, then w2 is paired with w3 , but not with w1 due to a unique 3-face incident with three 5-vertices at a poor vertex. R1. A 4+ -face f = v1 . . . vd(f ) gives each incident 5-vertex v2 : (a) 21 if d(v1 ) = d(v3 ) = 5, or (b) 34 if d(v1 ) ≥ 9 and d(v3 ) = 5. R2. A 5-vertex v with d(v1 ) ≥ 9 receives the following charge from its 9+ -neighbor v2 : 7 (a) if d(v3 ) = 5, then 12 in the case that d(v2 ) ≤ 12 and 12 otherwise, or 1 (b) 2 if d(v3 ) ≥ 9. R3. A non-simplicial 5-vertex v with d(v1 ) = d(v3 ) = d(v4 ) = 5 receives 41 from each of its 9+ -neighbors v2 and v5 . R4. A strong 5-vertex gives 16 to each of its two 5-neighbors. R5. A simplicial 5-vertex v with d(v1 ) = d(v2 ) = d(v4 ) = 5 receives from v5 : (a) 31 if 9 ≤ d(v5 ) ≤ 10, and (b) 12 if 11 ≤ d(v5 ) ≤ 12. R6. If a simplicial vertex v satisfies d(v1 ) = d(v2 ) = d(v4 ) = 5 and d(v5 ) ≥ 13, then v5 gives 21 to v , with the following two exceptions: 7 (ex1) if 9 ≤ d(v3 ) ≤ 10 and v2 has three 5-neighbors, then v5 gives 12 to v ; 7 (ex2) if d(v3 ) ≥ 13, v is paired with a poor vertex v1 , and v2 has three 5-neighbors, then v5 gives 12 to v . 1 R7. A poor 5-vertex v receives 12 from its paired vertex v1 if v2 has three 5-neighbors. 2.2. Checking µ′ (x) ≥ 0 whenever x ∈ V ∪ F Note that a 3-face f does not participate in discharging, so µ′ (f ) = µ(f ) = 2 × 3 − 6 = 0. Now if f is a 4+ -face, then its donation of 34 by R1b may be interpreted as giving 12 to the 5-vertex and 41 to the neighbor 9+ -vertex along the boundary 3(d(f )−4) of f . As a result, each vertex incident with f receives at most 12 from f , so µ′ (f ) ≥ 2d(f ) − 6 − d(f ) × 21 = ≥ 0. 2 Now suppose v ∈ V . Case 1. d(v ) = 5. If v is adjacent to at least four 9+ -vertices, then µ′ (v ) ≥ 5 − 6 + 4 × 21 > 0 by R2. Suppose v has precisely three 9+ -neighbors. If they are consecutive, then v receives at least 3 × 21 from them by 1 R2 and can give 2 × 16 to the two 5-neighbors by R4 if v is strong, and 12 to the paired 5-vertex by R7, which implies
2442
O.V. Borodin, A.O. Ivanova and O.N. Kazak / Discrete Mathematics 342 (2019) 2439–2444
Fig. 1. Rules of discharging.
1 µ′ (v ) ≥ 3 × 21 − 2 × 61 − 12 > 0 in view of the one-to-one correspondence between poor and paired vertices. Now if d(v1 ) = d(v3 ) = 5, then v receives at least 12 from each of v4 and v5 by R2a. Since v is not strong, it gives no charge to v1 and v3 by R4. Also v cannot give charge by R7, since it has no consecutive 5-neighbors as we see in R7. So, we have µ′ (v ) ≥ −1 + 2 × 12 = 0. Since v cannot have four 5-neighbors, it remains to assume that v has precisely two 9+ -neighbors. First suppose d(v1 ) ≥ d(v2 ) ≥ 9. If d(v1 ) ≤ 12, then v receives 12 from each of v1 and v2 by R2a and gives nothing away by R7, so 1 7 to v by R2a. In turn, v gives to v5 either 12 by R7 if v is paired with v5 or no charge µ′ = 0. If d(v1 ) ≥ 13, then v1 gives 12 7 1 otherwise. In both cases, v1 brings at least 21 = 12 − 12 to v in total. The 9+ -vertex v2 also brings at least 12 to v by R2a possibly combined with R7, and hence µ′ (v ) ≥ −1 + 2 × 21 = 0. From now on we can assume that d(v1 ) ≥ 11 and d(v3 ) ≥ 9 due to the absence of (5, 5, 10, 5, 10)-vertices. If v is not simplicial, then v receives 2 × 41 from v1 and v3 by R3 and at least 21 from an incident 4+ -face by R1. We already have 1 µ′ (v ) ≥ 0 unless v gives 12 to at least one of v4 and v5 by R7. This happens only if the face f = · · · v4 vv5 is a triangle, in 1 which case v actually receives 43 by R1b from an incident 4+ -face, so µ′ (v ) ≥ −1 + 34 + 2 × 14 − 2 × 12 > 0. 1 Finally, suppose v is simplicial. If v gives 12 to v5 by R7, so that v is paired with a poor vertex v5 , then d(v1 ) ≥ 13 and v4 is not strong. The latter implies that v4 has a 5-neighbor different from v and v5 . In turn, this means that d(v3 ) ≥ 13 since otherwise v would be a (5, 5, 10, 5, 12)-vertex, a contradiction. 7 7 1 Therefore, v receives 12 from v1 by R6ex2, and hence v1 brings 12 − 12 to v . We note that v3 also brings 12 to v either 1 ′ by R6 alone or by R6ex2 combined with R7. Thus we have µ (v ) = −1 + 2 × 12 = 0 when v gives away 12 at least once
in R7.
O.V. Borodin, A.O. Ivanova and O.N. Kazak / Discrete Mathematics 342 (2019) 2439–2444
2443
1 From now we can assume that v is not a donator of 12 by R7. Each 11+ -neighbor gives v at least 12 by R5b and R6, so it remains to assume that d(v3 ) ≤ 10. This implies that d(v1 ) ≥ 13 due to the absence of (5, 5, 10, 5, 12)-vertices by assumption. 1 from v5 by R7 too), 13 from v3 by Thus our v is poor. If v5 is strong, then v receives 61 from v5 by R4 (and maybe 12 1 1 7 ′ R5a, and 2 from v1 by R6, which results in µ (v ) ≥ 0. Otherwise, v receives 12 from v5 by R7, 13 from v3 , and 12 from v1 by R6ex1, so µ′ (v ) = 0. Case 2. 9 ≤ d(v ) ≤ 10. To estimate the total donation of v by R2, R3 and R5a, we average donations of v to its 5-neighbors according to R2a,b as follows. Instead of giving 12 to a 5-neighbor v2 by R2b, let v give 61 to each of the 9+ -vertices v1 and v3 , so the modified donation to v2 becomes 61 . If d(v1 ) = d(v2 ) = 5 and d(v3 ) ≥ 9, as in R2a, then v instead can give 13 to v2 and 16 to the 9+ -vertex v3 . As a result, each neighbor receives at most 13 = 16 + 16 = 21 − 16 from v , so we see that µ′ (v ) ≥ d(v ) − 6 − d(3v) = 2(d(v3)−9) ≥ 0. d(v ) Case 3. 11 ≤ d(v ) ≤ 12. We note that v gives each neighbor at most 21 by R2, R3 and R5b, so µ′ (v ) ≥ d(v ) − 6 − 2 = d(v )−12 , which settles the case d(v ) = 12. 2 So suppose d(v ) = 11. If v has a 9+ -neighbor, then µ′ (v ) ≥ 11 − 6 − 10 × 12 = 0. Thus we can assume that v is completely surrounded by 5-vertices. If v is incident with a 4+ -face . . . v1 vv2 , then each of v1 and v2 receives nothing by R2 and at most 41 by R3 from v . This implies µ′ (v ) ≥ 5 − 2 × 14 − (11 − 2) × 21 = 0 in view of R5b. It remains to assume in addition that v is simplicial. Now if there is a 4+ -face . . . v1′ v1 v2 v2′ , then still each of v1 and v2 receives at most 41 from v by R3, so µ′ (v ) ≥ 0 as above. Thus we are done unless there are vertices w1 , . . . , w11 such that there are 3-faces wk vk vk+1 whenever 1 ≤ k ≤ 11 (addition modulo 11). If so, then we cannot have d(wk ) = d(wk+1 ) = 5 for any k due to the absence of (5, 5, 5, 5, ∞)vertices. By the oddness of 11, this implies that, say, d(w1 ) ≥ 9 and d(w2 ) ≥ 9, in which case v gives no charge to v2 by R2, R3 and R5b, which yields µ′ (v ) ≥ 5 − 10 × 12 = 0. 7 Case 4. d(v ) ≥ 13. We know that v gives at most 12 to each adjacent 5-vertex by R2, R3, and R6. Since µ(v ) = 7d(v ) 5d(v )−72 2 7 ′ d(v ) − 6 − 12 = , it follows that µ ( v ) > 0 for d(v ) ≥ 15, µ′ (v ) ≥ − 12 for d(v ) = 14, and µ′ (v ) ≥ − 12 12 for d(v ) = 13. Thus we are already done for d(v ) ≥ 15 and now we wish to improve these rough estimations for 13 ≤ d(v ) ≤ 14. 7(d(v )−1) We have nothing to prove unless v is completely surrounded by 5-vertices, for otherwise µ′ (v ) ≥ d(v ) − 6 − = 12 7(d(v )−2) 5(d(v )−13) 1 ′ ≥ 0. Also, the argument used in Case 3 shows that if v is not simplicial, then µ ( v ) ≥ d( v ) − 6 − 2 × − = 12 4 12 5(d(v )−13) 1 1 ′ ′ + ′ + > 0. Now if there is a 4 -face . . . v v v v , then we similarly have µ ( v ) ≥ due to R3. 1 2 1 2 12 12 12 As in Case 3, we now have a cyclic sequence Wd(v ) = w1 , . . . , wd(v ) such that there are 3-faces wk vk vk+1 (addition modulo d(v )) whenever 1 ≤ k ≤ d(v ). Again, there are no two consecutive 5-vertices in Wd(v ) since each vk must have a 9+ -neighbor other than v . If there are two consecutive 9+ -vertices in Wd(v ) , say w1 and w2 , then v2 receives no charge from v by R1–R7, so we 7 7 7 can improve our rough estimation µ′ (v ) ≥ − 12 to µ′ (v ) ≥ − 12 + 12 ≥ 0, as desired. Due to the oddness of 13, this completes the proof for d(v ) = 13. So suppose d(v ) = 14. If v has at least one non-simplicial 5-neighbor vk , then vk receives at most 14 from v by R3, so 7 ′ µ (v ) ≥ 8 − 14 − 13 × 12 > 0. Thus we can assume that v is completely surrounded by simplicial 5-vertices. This means that W14 is actually a 14-cycle, and we can assume that d(w1 ) = d(w3 ) = · · · = d(w13 ) = 5. Now if at least one of 5-vertices in W14 , say w1 , is strong, that is w1 has a 9+ -neighbor outside W14 , then each of v1 7 7 and v2 receives 12 by R6 rather than 12 by R6ex1 or R6ex2, which yields µ′ (v ) ≥ 8 − 2 × 12 − 12 × 12 = 0. Now all w1 , w3 , . . . , w13 are non-strong, that is each of them has three 5-neighbors. Among the seven 9+ -vertices w2 , w4 , . . . , w14 there are no two consecutive (cyclically) 10− -vertices, for otherwise we would have a (5, 5, 10, 5, 10)vertex, which is impossible. By parity reasons and symmetry, we can assume that d(w14 ) ≥ 11 and d(w2 ) ≥ 11. So each of v1 and v2 obeys the 7 general rule R6 rather than its exceptions R6ex1 or R6ex2. This means that again µ′ (v ) ≥ 8 − 2 × 21 − 12 × 12 = 0, as desired. Thus we have proved µ′ (x) ≥ 0 whenever x ∈ V ∪ F , which contradicts (1) and completes the proof of Theorem 3.
References [1] O.V. Borodin, A.O. Ivanova, Describing 4-stars at 5-vertices in normal plane maps with minimum degree 5, Discrete Math. 313 (17) (2013) 1710–1714. [2] O.V. Borodin, A.O. Ivanova, An analogue of franklin’s theorem, Discrete Math. 339 (10) (2016) 2553–2556. [3] O.V. Borodin, A.O. Ivanova, Light and low 5-stars in normal plane maps with minimum degree 5, Siberian Math. J. 57 (3) (2016) 470–475. [4] O.V. Borodin, A.O. Ivanova, Low 5-stars in normal plane maps with minimum degree 5, Discrete Math. 340 (2) (2017) 18–22. [5] O.V. Borodin, A.O. Ivanova, New results about the structure of plane graphs: a survey, AIP Conf. Proc. 1907 (2017) 030051. [6] O.V. Borodin, A.O. Ivanova, On light neighborhoods of 5-vertices in 3-polytopes with minimum degree 5, Discrete Math. 340 (9) (2017) 2234–2242. [7] O.V. Borodin, A.O. Ivanova, T.R. Jensen, 5-stars of low weight in normal plane maps with minimum degree 5, Discuss. Math. Graph Theory 34 (3) (2014) 539–546. [8] O.V. Borodin, A.O. Ivanova, O.N. Kazak, Describing neighborhoods of 5-vertices in 3-polytopes with minimum degree 5 and without vertices of degrees from 7 to 11, Discuss. Math. Graph Theory 38 (3) (2018) 615–625.
2444
O.V. Borodin, A.O. Ivanova and O.N. Kazak / Discrete Mathematics 342 (2019) 2439–2444
[9] O.V. Borodin, A.O. Ivanova, O.N. Kazak, E.I. Vasil’eva, Heights of minor 5-stars in 3-polytopes with minimum degree 5 and no vertices of degree 6 and 7, Discrete Math. 341 (3) (2018) 825–829. [10] O.V. Borodin, A.O. Ivanova, D.V. Nikiforov, Low and light 5-stars in 3-polytopes with minimum degree 5 and restrictions on the degrees of major vertices, Siberian Math. J. 58 (4) (2017) 600–605. [11] O.V. Borodin, A.O. Ivanova, D.V. Nikiforov, Low minor 5-stars in 3-polytopes with minimum degree 5 and no 6-vertices, Discrete Math. 340 (7) (2017) 1612–1616. [12] O.V. Borodin, D.R. Woodall, Short cycles of low weight in normal plane maps with minimum degree 5, Discuss. Math. Graph Theory 18 (2) (1998) 159–164. [13] B. Ferencová, T. Madaras, Light graph in families of polyhedral graphs with prescribed minimum degree, face size, edge and dual edge weight, Discrete Math. 310 (12) (2010) 1661–1675. [14] J. Harant, S. Jendrol’, On the existence of specific stars in planar graphs, Graphs Combin. 23 (2007) 529–543. [15] A.O. Ivanova, D.V. Nikiforov, The structure of neighborhoods of 5-vertices in plane triangulation with minimum degree 5 (Russian), Math. Notes Yakutsk State Univ. 20 (2) (2013) 66–78. [16] A.O. Ivanova, D.V. Nikiforov, Combinatorial structure of triangulated 3-polytopes with minimum degree 5 (Russian), in: Proceedings of the Scientific Conference of Students, Graduate Students, and Young Researchers. XVII and XVIII Lavrent’ev’s Reading, International Center for Research Projects, Yakutsk; Kirov, 2015, pp. 22–27. [17] S. Jendrol’, A structural property of convex 3-polytopes, Geom. Dedicata 68 (1997) 91–99. [18] S. Jendrol’, T. Madaras, On light subgraphs in plane graphs of minimal degree five, Discuss. Math. Graph Theory 16 (1996) 207–217. [19] S. Jendrol’, T. Madaras, Note on an existence of small degree vertices with at most one big degree neighbour in planar graphs, Tatra Mt. Math. Publ. 30 (2005) 149–153. [20] S. Jendrol’, H.-J. Voss, Light subgraphs of graphs embedded in the plane — a survey, Discrete Math. 313 (4) (2013) 406–421. [21] H. Lebesgue, Quelques conséquences simples de la formule d’Euler, J. Math. Pures Appl. 19 (1940) 27–43. [22] D.V. Nikiforov, The structure of neighborhoods of normal plane maps with minimum degree 5 (Russian), Math. Notes North-Eastern Federal Univ. 23 (1) (2016) 56–66. [23] Ph. Franklin, The four colour problem, Amer. J. Math. 44 (1922) 225–236. [24] P. Wernicke, Über den kartographischen Vierfarbensatz, Math. Ann. 58 (1904) 413–426.