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An analogue of Franklin’s Theorem
Discrete Mathematics 339 (2016) 2553–2556
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Note
An analogue of Franklin’s Theorem O.V. Borodin a , A.O. Ivanova b,∗ a
Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia
b
Ammosov North-Eastern Federal University, Yakutsk, 677000, Russia
article
info
Article history: Received 8 September 2015 Received in revised form 29 April 2016 Accepted 30 April 2016
Keywords: Planar graph Plane map Structure properties 3-polytope Weight
abstract Back in 1922, Franklin proved that every 3-polytope P5 with minimum degree 5 has a 5-vertex adjacent to two vertices of degree at most 6, which is tight. This result has been extended and refined in several directions. The purpose of this note is to prove that every P5 has a vertex of degree at most 6 adjacent to a 5-vertex and another vertex of degree at most 6, which is also tight. Moreover, we prove that there is no tight description of 3-paths in P5 s other than these two. © 2016 Elsevier B.V. All rights reserved.
1. Introduction The degree d(x) of a vertex or face x in a plane graph G is the number of its incident edges. A k-vertex (k-face) is a vertex (face) with degree k, a k+ -vertex has degree at least k, etc. The minimum vertex degree of G is δ(G). We will drop the arguments whenever this does not lead to confusion. A k-path is a path on k vertices. A path uvw is an (i, j, k)-path if d(u) ≤ i, d(v) ≤ j, and d(w) ≤ k. The weight w(H ) of a subgraph H of a graph G is the degree-sum of the vertices of H in G. By Pδ denote the class of 3-polytopes with minimum degree δ ; in particular, P3 is the set of all 3-polytopes. In 1904, Wernicke [35] proved that if P5 ∈ P5 then P5 contains a 5-vertex adjacent to a 6− -vertex. This result was strengthened by Franklin [19] in 1922 by proving the existence of a (6, 5, 6)-path in every P5 . Theorem 1 (Franklin [19]). Every 3-polytope with minimum degree 5 has a (6, 5, 6)-path, which is tight. We recall that a description of 3-paths is tight if none of its parameters can be strengthened and no term dropped. The tightness of Franklin’s description is shown by putting a vertex inside each face of the dodecahedron and joining it to the five boundary vertices. Franklin’s Theorem 1 is fundamental in the structural theory of planar graphs; it has been extended or refined in several directions, see [1–18,20–24,26–34] and a survey Jendrol’–Voss [25]. We now mention only a few easily formulated results on P5 , which are the closest to Franklin’s Theorem and whose parameters are all sharp. Borodin [3] proved that there is a 3-face with weight at most 17. Jendrol’ and Madaras [23] ensured a 5-vertex that has three neighbors whose weight sums to at most 18 and a 4-path with weight at most 23. Madaras [29] found a 5-path with weight at most 29.
∗
Corresponding author. E-mail addresses: [email protected] (O.V. Borodin), [email protected] (A.O. Ivanova).
http://dx.doi.org/10.1016/j.disc.2016.04.019 0012-365X/© 2016 Elsevier B.V. All rights reserved.
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O.V. Borodin, A.O. Ivanova / Discrete Mathematics 339 (2016) 2553–2556
Fig. 1. A construction showing the tightness of Theorem 2.
All of a sudden, we have realized that the following fact, maybe the most similar to Franklin’s Theorem, is missing in the literature. Theorem 2. Every 3-polytope with minimum degree 5 has a (5, 6, 6)-path, which is tight. Recently, we proved [11] that there exist precisely seven tight descriptions of 3-paths in triangle-free 3-polytopes. Theorem 3 (Borodin and Ivanova [11]). There exist precisely seven tight descriptions of 3-paths in triangle-free 3-polytopes: (i) (5, 3, 6) ∨ (4, 3, 7), (ii) (3, 5, 3) ∨ (3, 4, 4), (iii) (5, 3, 6) ∨ (3, 4, 3), (iv) (3, 5, 3) ∨ (4, 3, 4), (v) (5, 3, 7), (vi) (3, 5, 4), (vii) (5, 4, 6). Problem 4 (Borodin, Ivanova, and Kostochka [16]). Describe all tight descriptions of 3-paths in P3 . Another purpose of our short note is to make the following modest contribution to Problem 4. Theorem 5. There are no tight descriptions of 3-paths in P5 s other than those given by Franklin’s Theorem and Theorem 2. 2. Proving Theorem 2 To show the tightness of Theorem 2, it suffices to replace each face of the icosahedron by the configuration shown in Fig. 1. Indeed, the resulting graph H2 has neither (5, 6, 5)-paths nor (5, 5, 6)-paths. Now suppose that a 3-polytope P5′ contradicts Theorem 2 by avoiding (5, 6, 6)-paths. Let P5 be a counterexample to Theorem 2 on the same vertices as P5′ having the most edges. Let v1 , . . . , vd(x) denote the neighbors of a vertex or a face x in a cyclic order round x. (*) P5 is a triangulation. Indeed, suppose P5 has a 4+ -face f = v1 , . . . , vd(f ) . If d(v1 ) ≥ 6 or d(v3 ) ≥ 6, then adding the diagonal d = v1 v3 results in a counterexample P5∗ to Theorem 2 with more edges since d joins in P5∗ two 7+ -vertices, which contradicts the definition of P5 . Thus d(v1 ) = d(v3 ) = 5 in P5 . By symmetry, we have also d(v2 ) = d(v4 ) = 5. This means that P5 has a (5, 5, 5)-paths, a contradiction. Denote the sets of vertices, edges, and faces of P5 by V , E and F , respectively. Euler’s formula |V | − |E | + |F | = 2 for P5 yields
(d(v) − 6) = −12.
(1)
v∈V
We assign an initial charge µ(v) = d(v) − 6 to each v ∈ V . Note that only 5-vertices have a negative initial charge. Using the properties of G as a counterexample to Theorem 2, we will define a local redistribution of charges, preserving their sum, such that the new charge µ′ (v) is non-negative whenever v ∈ V . This will contradict the fact that the sum of the new charges is, by (1), equal to −12, and this contradiction will finish the proof of Theorem 2. Namely, we use the following discharging rules. R1. Every 6+ -vertex gives +
R2. Every 7 -vertex gives
1 4 1 8
to every adjacent 5-vertex. to every adjacent 6-vertex.
O.V. Borodin, A.O. Ivanova / Discrete Mathematics 339 (2016) 2553–2556
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We now check that µ′ (v) ≥ 0 whenever v ∈ V . CASE 1. d(v) = 5. Since v has at most one 5-neighbor due to the absence of (5, 5, 5)-paths in our triangulation P5 , it follows by R1 that µ′ (v) ≥ −1 + 4 × 41 = 0.
CASE 2. d(v) = 6. If v has a 5-neighbor v1 , then d(vi ) ≥ 7 whenever 2 ≤ i ≤ 6 due to the absence of (5, 6, 6)-paths, which implies by R1, R2 that µ′ (v) ≥ 0 − 14 + 5 × 18 > 0. Otherwise, µ′ (v) ≥ µ(v) = 0
CASE 3. d(v) = 7. If v has at most one 5-neighbor, then µ′ (v) ≥ 1 − 14 − (7 − 1) × 18 = 0 by R1, R2 combined with (*). Now suppose that v has precisely two 5-neighbors; then v has at most four 6-neighbors due to (*) and the absence of (5, 6, 6)-paths around v . This implies µ′ (v) ≥ 1 − 2 × 14 − 4 × 18 = 0. We next show that µ′ (v) ≥ 0 if there are two consecutive 5-vertices round v , say d(v1 ) = d(v2 ) = 5. Indeed, this implies d(v3 ) ≥ 7 and d(v7 ) ≥ 7 by (*). It remains to observe that among v4 , v5 , and v6 either there is a 7+ -vertex, in which case µ′ (v) ≥ 1 − 4 × 41 = 0 by R1, or at least two 6-vertices, in which case we have µ′ (v) ≥ 1 − 3 × 14 − 2 × 18 = 0. The last possibility to consider is d(v1 ) = d(v3 ) = d(v5 ) = 5, d(v2 ) ≥ 7, d(v4 ) ≥ 7, d(v6 ) ≥ 6, and d(v7 ) ≥ 6. Here, we have µ′ (v) ≥ 1 − 3 × 14 − 2 × 81 = 0. 3(d(v)−8) 4
CASE 4. d(v) ≥ 8. By R1, R2, we have µ′ (v) ≥ d(v) − 6 − d(v) × 41 = A contradiction 0 ≤ −12 with (1) completes the proof of Theorem 2.
≥ 0, as desired.
3. Proving Theorem 5 Suppose D = x1 y1 z1 ∨ · · · ∨ xk yk zk is a tight description of 3-paths in P5 . This means that (1) every P5 ∈ P5 has a (xi , yi , zi )-path for at least one i with 1 ≤ i ≤ k, and (2) if we delete any term xi yi zi from D or decrease any parameter in D by one without changing the other 3k − 1 parameters, then the new description is not satisfied by at least one P5 ∈ P5 . Note that, due to its tightness, the description D cannot have triplets XYZ and X ′ Y ′ Z ′ such that X ≤ X ′ , Y ≤ Y ′ , and Z ≤ Z ′ , for D′ = D \ {XYZ } is equivalent to D but shorter. Also, all parameters in D should be at least 5 since we deal with P5 . CASE 1. D has a term XYZ = 5+ 6+ 6+ . By Theorem 2, D is true and not stronger than the tight description 566, which implies that D = 566. CASE 2. D has a term XYZ = 6+ 5+ 6+ . By Theorem 1, D is true and not stronger than the tight description 656, so D = 656. CASE 3. Every term in D has at most one parameter greater than 5. Such a D is not a description at all, since all 3-paths in the graph H partly shown in Fig. 1 go through at most one 5-vertex. This proves that there are precisely two tight descriptions of 3-paths in P5 , as desired. Acknowledgments The first author was supported by the Russian Foundation for Basic Research (grants 15-01-05867 and 16-01-00499) and by President Grants for Government Support of the Leading Scientific Schools of the Russian Federation (grant NSh1939.2014.1). The second author’s work was performed as a part of government work ‘‘Organizing research’’. References [1] V.A. Aksenov, O.V. Borodin, A.O. Ivanova, Weight of 3-paths in sparse plane graphs, Electron. J. Combin. 22 (3) (2015) Paper #P3.28. [2] K. Ando, S. Iwasaki, A. Kaneko, Every 3-connected planar graph has a connected subgraph with small degree sum, in: Annual Meeting of Mathematical Society of Japan, 1993 (in Japanese). [3] O.V. Borodin, Solution of Kotzig’s and Grünbaum’s problems on the separability of a cycle in a planar graph, Mat. Zametki 46 (5) (1989) 9–12 (in Russian). [4] O.V. Borodin, Structural properties of plane maps with minimum degree 5, Math. Nachr. 18 (1992) 109–117. [5] O.V. Borodin, Structural theorem on plane graphs with application to the entire coloring, J. Graph Theory 23 (3) (1996) 233–239. [6] O.V. Borodin, Minimal vertex degree sum of a 3-path in plane maps, Discuss. Math. Graph Theory 17 (2) (1997) 279–284. [7] O.V. Borodin, A.O. Ivanova, Describing (d − 2)-stars at d-vertices, d ≤ 5, in normal plane maps, Discrete Math. 313 (17) (2013) 1700–1709. [8] 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. [9] O.V. Borodin, A.O. Ivanova, Describing 3-faces in normal plane maps with minimum degree 4, Discrete Math. 313 (23) (2013) 2841–2847. [10] O.V. Borodin, A.O. Ivanova, Each 3-polytope with minimum degree 5 has a 7-cycle with maximum degree at most 15, Sibirsk. Mat. Zh. 56 (4) (2015) 775–789 (in Russian). [11] O.V. Borodin, A.O. Ivanova, Describing tight descriptions of 3-paths in triangle-free normal plane maps, Discrete Math. 338 (2015) 1947–1952. [12] 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. [13] O.V. Borodin, A.O. Ivanova, T.R. Jensen, A.V. Kostochka, M.P. Yancey, Describing 3-paths in normal plane maps, Discrete Math. 313 (23) (2013) 2702–2711. [14] O.V. Borodin, A.O. Ivanova, A.V. Kostochka, Every 3-polytope with minimum degree 5 has a 6-cycle with maximum degree at most 11, Discrete. Math. 315–316 (2014) 128–134. [15] O.V. Borodin, A.O. Ivanova, A.V. Kostochka, Describing faces in plane triangulations, Discrete. Math. 319 (2014) 47–61. [16] O.V. Borodin, A.O. Ivanova, A.V. Kostochka, Tight descriptions of 3-paths in normal plane maps, J. Graph Theory (in press).
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[17] 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. [18] B. Ferencová, T. Madaras, Light graphs in families of polyhedral graphs with prescribed minimum degree, face size, edge and dual edge weight, Discrete Math. 310 (2010) 1661–1675. [19] P. Franklin, The four color problem, Amer. J. Math. 44 (1922) 225–236. [20] P. Hudak, T. Madaras, On doubly light triangles in plane graphs, Discrete Math. 313 (19) (2013) 1978–1988. [21] S. Jendrol’, A structural property of convex 3-polytopes, Geom. Dedicata 68 (1997) 91–99. [22] S. Jendrol’, Paths with restricted degrees of their vertices in planar graphs, Czechoslovak Math. J. 49 (124) (1999) 481–490. [23] S. Jendrol’, T. Madaras, On light subgraphs in plane graphs with minimum degree five, Discuss. Math. Graph Theory 16 (1996) 207–217. [24] S. Jendrol’, T. Madaras, R. Soták, Z. Tuza, On light cycles in plane triangulations, Discrete Math. 197/198 (1999) 453–467. [25] S. Jendrol’, H.-J. Voss, Light subgraphs of graphs embedded in the plane—a survey, Discrete Math. 313 (2013) 406–421. [26] S. Jendrol’, M. Maceková, Describing short paths in plane graphs of girth at least 5, Discrete Math. 338 (2015) 149–158. [27] S. Jendrol’, M. Maceková, R. Soták, Note on 3-paths in plane graphs of girth 4, Discrete Math. 338 (2015) 1643–1648. [28] H. Lebesgue, Quelques conséquences simples de la formule d’Euler, J. Math. Pures Appl. 19 (1940) 27–43. [29] T. Madaras, Note on the weight of paths in plane triangulations of minimum degree 4 and 5, Discuss. Math. Graph Theory 20 (2) (2000) 173–180. [30] T. Madaras, On the structure of plane graphs of minimum face size 5, Discuss. Math. Graph Theory 24 (3) (2004) 403–411. [31] T. Madaras, Two variations of Franklin’s theorem, Tatra Mt. Math. Publ. 36 (2007) 61–70. [32] T. Madaras, R. Škrekovski, H.-J. Voss, The 7-cycle C7 is light in the family of planar graphs with minimum degree 5, Discrete Math. 307 (2007) 1430–1435. [33] T. Madaras, R. Soták, The 10-cycle C10 is light in the family of all plane triangulations with minimum degree five, Tatra Mt. Math. Publ. 18 (1999) 35–56. [34] B. Mohar, R. Škrekovski, H.-J. Voss, Light subgraphs in planar graphs of minimum degree 4 and edge-degree 9, J. Graph Theory 44 (2003) 261–295. [35] 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
Note
An analogue of Franklin’s Theorem O.V. Borodin a , A.O. Ivanova b,∗ a
Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia
b
Ammosov North-Eastern Federal University, Yakutsk, 677000, Russia
article
info
Article history: Received 8 September 2015 Received in revised form 29 April 2016 Accepted 30 April 2016
Keywords: Planar graph Plane map Structure properties 3-polytope Weight
abstract Back in 1922, Franklin proved that every 3-polytope P5 with minimum degree 5 has a 5-vertex adjacent to two vertices of degree at most 6, which is tight. This result has been extended and refined in several directions. The purpose of this note is to prove that every P5 has a vertex of degree at most 6 adjacent to a 5-vertex and another vertex of degree at most 6, which is also tight. Moreover, we prove that there is no tight description of 3-paths in P5 s other than these two. © 2016 Elsevier B.V. All rights reserved.
1. Introduction The degree d(x) of a vertex or face x in a plane graph G is the number of its incident edges. A k-vertex (k-face) is a vertex (face) with degree k, a k+ -vertex has degree at least k, etc. The minimum vertex degree of G is δ(G). We will drop the arguments whenever this does not lead to confusion. A k-path is a path on k vertices. A path uvw is an (i, j, k)-path if d(u) ≤ i, d(v) ≤ j, and d(w) ≤ k. The weight w(H ) of a subgraph H of a graph G is the degree-sum of the vertices of H in G. By Pδ denote the class of 3-polytopes with minimum degree δ ; in particular, P3 is the set of all 3-polytopes. In 1904, Wernicke [35] proved that if P5 ∈ P5 then P5 contains a 5-vertex adjacent to a 6− -vertex. This result was strengthened by Franklin [19] in 1922 by proving the existence of a (6, 5, 6)-path in every P5 . Theorem 1 (Franklin [19]). Every 3-polytope with minimum degree 5 has a (6, 5, 6)-path, which is tight. We recall that a description of 3-paths is tight if none of its parameters can be strengthened and no term dropped. The tightness of Franklin’s description is shown by putting a vertex inside each face of the dodecahedron and joining it to the five boundary vertices. Franklin’s Theorem 1 is fundamental in the structural theory of planar graphs; it has been extended or refined in several directions, see [1–18,20–24,26–34] and a survey Jendrol’–Voss [25]. We now mention only a few easily formulated results on P5 , which are the closest to Franklin’s Theorem and whose parameters are all sharp. Borodin [3] proved that there is a 3-face with weight at most 17. Jendrol’ and Madaras [23] ensured a 5-vertex that has three neighbors whose weight sums to at most 18 and a 4-path with weight at most 23. Madaras [29] found a 5-path with weight at most 29.
∗
Corresponding author. E-mail addresses: [email protected] (O.V. Borodin), [email protected] (A.O. Ivanova).
http://dx.doi.org/10.1016/j.disc.2016.04.019 0012-365X/© 2016 Elsevier B.V. All rights reserved.
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Fig. 1. A construction showing the tightness of Theorem 2.
All of a sudden, we have realized that the following fact, maybe the most similar to Franklin’s Theorem, is missing in the literature. Theorem 2. Every 3-polytope with minimum degree 5 has a (5, 6, 6)-path, which is tight. Recently, we proved [11] that there exist precisely seven tight descriptions of 3-paths in triangle-free 3-polytopes. Theorem 3 (Borodin and Ivanova [11]). There exist precisely seven tight descriptions of 3-paths in triangle-free 3-polytopes: (i) (5, 3, 6) ∨ (4, 3, 7), (ii) (3, 5, 3) ∨ (3, 4, 4), (iii) (5, 3, 6) ∨ (3, 4, 3), (iv) (3, 5, 3) ∨ (4, 3, 4), (v) (5, 3, 7), (vi) (3, 5, 4), (vii) (5, 4, 6). Problem 4 (Borodin, Ivanova, and Kostochka [16]). Describe all tight descriptions of 3-paths in P3 . Another purpose of our short note is to make the following modest contribution to Problem 4. Theorem 5. There are no tight descriptions of 3-paths in P5 s other than those given by Franklin’s Theorem and Theorem 2. 2. Proving Theorem 2 To show the tightness of Theorem 2, it suffices to replace each face of the icosahedron by the configuration shown in Fig. 1. Indeed, the resulting graph H2 has neither (5, 6, 5)-paths nor (5, 5, 6)-paths. Now suppose that a 3-polytope P5′ contradicts Theorem 2 by avoiding (5, 6, 6)-paths. Let P5 be a counterexample to Theorem 2 on the same vertices as P5′ having the most edges. Let v1 , . . . , vd(x) denote the neighbors of a vertex or a face x in a cyclic order round x. (*) P5 is a triangulation. Indeed, suppose P5 has a 4+ -face f = v1 , . . . , vd(f ) . If d(v1 ) ≥ 6 or d(v3 ) ≥ 6, then adding the diagonal d = v1 v3 results in a counterexample P5∗ to Theorem 2 with more edges since d joins in P5∗ two 7+ -vertices, which contradicts the definition of P5 . Thus d(v1 ) = d(v3 ) = 5 in P5 . By symmetry, we have also d(v2 ) = d(v4 ) = 5. This means that P5 has a (5, 5, 5)-paths, a contradiction. Denote the sets of vertices, edges, and faces of P5 by V , E and F , respectively. Euler’s formula |V | − |E | + |F | = 2 for P5 yields
(d(v) − 6) = −12.
(1)
v∈V
We assign an initial charge µ(v) = d(v) − 6 to each v ∈ V . Note that only 5-vertices have a negative initial charge. Using the properties of G as a counterexample to Theorem 2, we will define a local redistribution of charges, preserving their sum, such that the new charge µ′ (v) is non-negative whenever v ∈ V . This will contradict the fact that the sum of the new charges is, by (1), equal to −12, and this contradiction will finish the proof of Theorem 2. Namely, we use the following discharging rules. R1. Every 6+ -vertex gives +
R2. Every 7 -vertex gives
1 4 1 8
to every adjacent 5-vertex. to every adjacent 6-vertex.
O.V. Borodin, A.O. Ivanova / Discrete Mathematics 339 (2016) 2553–2556
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We now check that µ′ (v) ≥ 0 whenever v ∈ V . CASE 1. d(v) = 5. Since v has at most one 5-neighbor due to the absence of (5, 5, 5)-paths in our triangulation P5 , it follows by R1 that µ′ (v) ≥ −1 + 4 × 41 = 0.
CASE 2. d(v) = 6. If v has a 5-neighbor v1 , then d(vi ) ≥ 7 whenever 2 ≤ i ≤ 6 due to the absence of (5, 6, 6)-paths, which implies by R1, R2 that µ′ (v) ≥ 0 − 14 + 5 × 18 > 0. Otherwise, µ′ (v) ≥ µ(v) = 0
CASE 3. d(v) = 7. If v has at most one 5-neighbor, then µ′ (v) ≥ 1 − 14 − (7 − 1) × 18 = 0 by R1, R2 combined with (*). Now suppose that v has precisely two 5-neighbors; then v has at most four 6-neighbors due to (*) and the absence of (5, 6, 6)-paths around v . This implies µ′ (v) ≥ 1 − 2 × 14 − 4 × 18 = 0. We next show that µ′ (v) ≥ 0 if there are two consecutive 5-vertices round v , say d(v1 ) = d(v2 ) = 5. Indeed, this implies d(v3 ) ≥ 7 and d(v7 ) ≥ 7 by (*). It remains to observe that among v4 , v5 , and v6 either there is a 7+ -vertex, in which case µ′ (v) ≥ 1 − 4 × 41 = 0 by R1, or at least two 6-vertices, in which case we have µ′ (v) ≥ 1 − 3 × 14 − 2 × 18 = 0. The last possibility to consider is d(v1 ) = d(v3 ) = d(v5 ) = 5, d(v2 ) ≥ 7, d(v4 ) ≥ 7, d(v6 ) ≥ 6, and d(v7 ) ≥ 6. Here, we have µ′ (v) ≥ 1 − 3 × 14 − 2 × 81 = 0. 3(d(v)−8) 4
CASE 4. d(v) ≥ 8. By R1, R2, we have µ′ (v) ≥ d(v) − 6 − d(v) × 41 = A contradiction 0 ≤ −12 with (1) completes the proof of Theorem 2.
≥ 0, as desired.
3. Proving Theorem 5 Suppose D = x1 y1 z1 ∨ · · · ∨ xk yk zk is a tight description of 3-paths in P5 . This means that (1) every P5 ∈ P5 has a (xi , yi , zi )-path for at least one i with 1 ≤ i ≤ k, and (2) if we delete any term xi yi zi from D or decrease any parameter in D by one without changing the other 3k − 1 parameters, then the new description is not satisfied by at least one P5 ∈ P5 . Note that, due to its tightness, the description D cannot have triplets XYZ and X ′ Y ′ Z ′ such that X ≤ X ′ , Y ≤ Y ′ , and Z ≤ Z ′ , for D′ = D \ {XYZ } is equivalent to D but shorter. Also, all parameters in D should be at least 5 since we deal with P5 . CASE 1. D has a term XYZ = 5+ 6+ 6+ . By Theorem 2, D is true and not stronger than the tight description 566, which implies that D = 566. CASE 2. D has a term XYZ = 6+ 5+ 6+ . By Theorem 1, D is true and not stronger than the tight description 656, so D = 656. CASE 3. Every term in D has at most one parameter greater than 5. Such a D is not a description at all, since all 3-paths in the graph H partly shown in Fig. 1 go through at most one 5-vertex. This proves that there are precisely two tight descriptions of 3-paths in P5 , as desired. Acknowledgments The first author was supported by the Russian Foundation for Basic Research (grants 15-01-05867 and 16-01-00499) and by President Grants for Government Support of the Leading Scientific Schools of the Russian Federation (grant NSh1939.2014.1). The second author’s work was performed as a part of government work ‘‘Organizing research’’. References [1] V.A. Aksenov, O.V. Borodin, A.O. Ivanova, Weight of 3-paths in sparse plane graphs, Electron. J. Combin. 22 (3) (2015) Paper #P3.28. [2] K. Ando, S. Iwasaki, A. Kaneko, Every 3-connected planar graph has a connected subgraph with small degree sum, in: Annual Meeting of Mathematical Society of Japan, 1993 (in Japanese). [3] O.V. Borodin, Solution of Kotzig’s and Grünbaum’s problems on the separability of a cycle in a planar graph, Mat. Zametki 46 (5) (1989) 9–12 (in Russian). [4] O.V. Borodin, Structural properties of plane maps with minimum degree 5, Math. Nachr. 18 (1992) 109–117. [5] O.V. Borodin, Structural theorem on plane graphs with application to the entire coloring, J. Graph Theory 23 (3) (1996) 233–239. [6] O.V. Borodin, Minimal vertex degree sum of a 3-path in plane maps, Discuss. Math. Graph Theory 17 (2) (1997) 279–284. [7] O.V. Borodin, A.O. Ivanova, Describing (d − 2)-stars at d-vertices, d ≤ 5, in normal plane maps, Discrete Math. 313 (17) (2013) 1700–1709. [8] 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. [9] O.V. Borodin, A.O. Ivanova, Describing 3-faces in normal plane maps with minimum degree 4, Discrete Math. 313 (23) (2013) 2841–2847. [10] O.V. Borodin, A.O. Ivanova, Each 3-polytope with minimum degree 5 has a 7-cycle with maximum degree at most 15, Sibirsk. Mat. Zh. 56 (4) (2015) 775–789 (in Russian). [11] O.V. Borodin, A.O. Ivanova, Describing tight descriptions of 3-paths in triangle-free normal plane maps, Discrete Math. 338 (2015) 1947–1952. [12] 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. [13] O.V. Borodin, A.O. Ivanova, T.R. Jensen, A.V. Kostochka, M.P. Yancey, Describing 3-paths in normal plane maps, Discrete Math. 313 (23) (2013) 2702–2711. [14] O.V. Borodin, A.O. Ivanova, A.V. Kostochka, Every 3-polytope with minimum degree 5 has a 6-cycle with maximum degree at most 11, Discrete. Math. 315–316 (2014) 128–134. [15] O.V. Borodin, A.O. Ivanova, A.V. Kostochka, Describing faces in plane triangulations, Discrete. Math. 319 (2014) 47–61. [16] O.V. Borodin, A.O. Ivanova, A.V. Kostochka, Tight descriptions of 3-paths in normal plane maps, J. Graph Theory (in press).
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