Every Triangulated 3-polytope of Minimum Degree 4 Has a 4-path of Weight at Most 27

By δ and w k denote the minimum degree and minimum degree-sum (weight) of a k-vertex path in a given graph, respectively., where both bounds are sharp. For every 3-polytope with δ 4, we have sharp bounds w 2 11 (Lebesgue, 1940) and w 3 17 (Borodin, 1997). Madaras (2000) proved that every triangulated 3-polytope with δ 4 satisfies w 4 31 and constructed such a 3-polytope with w 4 = 27. We improve the Madaras bound w 4 31 to the sharp bound w 4 27.


Introduction
The degree of a vertex or face x in a graph G, that is the number of edges incident with x, is denoted by d(x).A k-vertex is a vertex v with d(v) = k.By k + or k − we denote any integer not smaller or not greater than k, respectively.Hence, a k + -face f satisfies d(f ) k, etc.
Let δ(G) be the minimum vertex degree, and w k (G) be the minimum degree sum (called weight) of a path on k vertices (k-path) in a graph G.
By G δ denote the class of plane graphs G with δ(G) δ; it is trivial that δ 5.The subset of 3-connected graphs in G δ is denoted by P δ .Given k, by w k denote the maximal weight of k-paths over G ∈ G δ or P ∈ P δ .
Already in 1904, Wernicke [40] proved that every P ∈ P 5 satisfies w 2 (P ) 11, which is tight.It follows from Lebesgue's [31] results of 1940 that P ∈ P 3 implies w 2 (G) 14, which was improved in 1955 by Kotzig [30] to the tight bound w 2 13.In 1972, Erdős (see [20]) conjectured that Kotzig's bound w 2 13 holds also in G 3 .Barnette (see [20]) announced to have proved this conjecture, but the proof has never appeared in print.The first published proof of Erdős' conjecture is due to Borodin [4].
A number of sharp upper bounds on w 2 have been obtained as lemmas in numerous papers on coloring sparse planar graphs (for a survey see Borodin [10]).A traditional measure of sparseness of a planar graph G is its girth g(G), which is the length of the shortest cycle in G. Another measure, suggested by Erdős (see [20]), is the absence of cycles of length from 4 to a certain constant.More generally, given a set S of integers, a graph is S-free if it has no cycle with length from S.
The first results of this type were known already to Lebesgue's [31] for P 3 : if g 4, then w 2 8, and if g 5, then w 2 = 6.
As for G 2 , we note that δ(K 2,t ) = 2 and w 2 (K 2,t ) = t + 2, so w 2 is unbounded in G 2 if g 4. In addition to forbidding certain collections S of cycle lengths, another way to find subclasses of G 2 with bounded w 2 is to impose restrictions on the set of 2-vertices in a graph.For example, forbidding 2-alternating cycles, which are cycles v 1 . . .v 2k with d(v 1 ) = d(v 3 ) = . . .= d(v 2k−1 ) = 2, we have w 2 15 (Borodin [4]).
The first application of this fact was to show that the total choosability of planar graphs with maximum degree ∆ at least 14 equals ∆+1 ( [4]).The notion of 2-alternating cycles, along with its more sophisticated analogues, have appeared in dozens of papers, since it sometimes provides crucial reducible configurations in coloring and partition problems (more often, on sparse graphs).
Some other results concerning the structure of edge neighborhoods in plane graphs can be found in [2, 5-7, 10, 14, 15, 18, 28, 29].So far, not many precise results have been obtained on w k with k 3. Back in 1922, Franklin [19] strengthened Wernicke's bound w 2 11 for P 5 in [40] to w 3 (G) 17.Both bounds 11 and 17 are sharp, as shown by putting a vertex inside each face of the dodecahedron and joining it with the five boundary vertices.
For 4-connected planar graphs with at least k vertices and any natural number k, Mohar [36] nicely proved a sharp bound w k 6k − 1, using Tutte's theorem [39] of 1956 that such graphs are hamiltonian.
We now consider sharp upper bounds on w 3 for graphs in P 2 with given girth g.It is an old folkloric fact that g(G) 16 implies w 3 (G) = 6, which probably first appeared in print in Nešetřil, Raspaud, and Sopena [38].
In 2000, Madaras [33] proved for triangulations with δ = 4 that w 4 31 and gave a construction with w 4 = 27 by putting a 3-cycle A 1 A 3 A 3 into each face Z 1 Z 2 Z 3 of the icosahedron followed by adding the edges A i Z j whenever 1 i, j 3 and i = j.
The purpose of our paper is to improve the bound w 4 31 by Madaras [33] to the sharp bound w 4 27.

Theorem 1. Every plane triangulation with δ
4 has a 4-path of weight at most 27, which is tight.

Proving Theorem 1
Proof of Theorem 1. Suppose that G is a counterexample to Theorem 1.In the course of the proof we should take care that a hypothetic 4-path P 4 = v 1 v 2 v 3 v 4 with w(P 4 ) 27 in G would not degenerate into a 3-cycle, which happens when v 1 coincides with v 4 .
A k-component c k is a maximal connected subgraph of G that consists of k vertices of degree at most 5. Clearly, any c 1 is simply a 5 − -vertex, c 2 consists of two adjacent vertices, and c 3 = uvw is either a 3-path (when there is no edge uw in G) or a 3-cycle.Furthermore, the following lemma shows, in particular, that if a 3-component c 3 is a cycle, then it is the boundary of a 3-face.Proof of Lemma 2. Suppose G has a separating 3-cycle S = uvw, which means that at least one vertex of G lies inside S and at least one outside S. Note that u has at least one neighbor inside S and at least one outside S, for otherwise {v, w} is a separating set in G, contrary to the 3-connectedness of G.The same is true for v and w.
First suppose that S consists of 5 − -vertices, which implies that at most nine edges join S to vertices not on S. By symmetry, we can assume that at most four edges lead from S inside (rather than outside) S. Again by symmetry, we can assume that each of u and v has precisely one neighbor inside S. Since G is a triangulation, there are 3-faces uvx, vwx, and wxu inside S.This implies that there is precisely one vertex x, necessarily with d(x) = 3, inside S; a contradiction.Now if d(u) = d(v) = 4, then the same argument works, which proves the second part of Lemma 2. Lemma 3. G has no k-component with k 4.
Proof of Lemma 3. Suppose on the contrary that there is a connected subgraph H of G on four 5 − -vertices.Since G has no 4-path P 4 with w(P 4 ) 4 × 5, it follows that H does not contain a 4-path.In turn, this means that H is a star, with three rays and a 5 − vertex as a center.However, the center forms a 3-face with two consecutive neighbors in our triangulation G, which produces a 4-path on H, a contradiction.

Discharging
where V , E, and F are sets of vertices, edges, and faces of G, respectively.By V 6 + and V 5 − denote the sets of 6 + -vertices and 5 − -vertices in V , respectively, so (1) can be written as follows.
By M C denote the set of minor components [mc] of G.It follows from Lemma 3 that V 5 − is split into minor components, each of which is a k − -component with k 3.
Every vertex v contributes the initial charge µ(v) = d(v) − 6 to (2), so only the charges of 5 − -vertices are negative.The initial charge of a minor component [mc] is the total initial charge of its vertices.Thus (2) can be rewritten as follows.
Using the properties of M as a counterexample, we define a local redistribution of µ's, preserving their sum, such that the new charge µ is non-negative for all v ∈ V 6 + and [mc] ∈ M C.This will contradict the fact that the sum of the new charges is, by (3), equal to −12.
For an integer k with k 6, we put ψ(k) = k−6 k .Our rules of discharging are as follows (see Fig. 6, and d(x) 6, the vertex u receives ξ(v) and ξ(x) through f from v and x, respectively.R2.For each 3-face f = xuv with d(u) 5, d(v) 5, and d(x) 6, each of u and v receives ξ(x) 2 from x through f , with the following exception: and there is a 3-face uvw with d(w) = 4, then (the 16 + -vertex) x gives 1  2 to each of u and v through f .R3.For each 3-face f = xuv with d(u) = d(v) = 4 and 6 d(x) 7, each of u and v receives 1  4 from x through f .In turn, x receives 1 12 from each adjacent 13 + -vertex y through each 3-face xyz such that d(z) 13.
Concerning the second part of R3, we note that there are at least three 3-faces incident with x and two 13 + -vertices, so x receives at least 6 × 1  12 through such faces.
6 ∨ 7 14 + ∨ 13 + 14 + ∨ 13 + 14 + ∨ 13 + 14 + ∨ 13 + q q q s % E W X y Here, u and v have neighbors z 1 , . . ., z 5 shown in Fig. 2b, and c 2 receives charge from Z by R1 and R2 through twelve faces.However, now Lemma 2 is not applied, thus the edge uv can belong to a separating 3-cycle z 2 uv, which is only possible when z 2 ∈ {z 4 , z 5 }.If so, then z 2 appears twice in Z and can have a relatively small degree, say 6 or 7, with no contradiction (since we are looking for a light 4-path rather than a light 3-cycle).Still, c 2 has at least three neighbors other than z k .
This time, we can ignore the donations from z k , since each neighbor other than z k has degree at least 19 − d(z k ) 10. Indeed, otherwise we would have a 4-path of weight at most 9 + 4 + 5 + 9 on {z k , u, v, z l } with z l = z k , which is impossible.Therefore, c 2 has at least three 10 + -neighbors, and each of them gives at least 2 5 to c 2 through each incident face by R1 and R2.Altogether, such donations happen at least eight times, which results in µ (c 2 ) 4 − 6 + 5 − 6 + 8 × 2 5 > 0. Case 3. d(u) = d(v) = 5.Now u and v have six neighbors z 1 , . . ., z 6 shown in Fig. 2c.Again Lemma 2 is not applied, and it can happen that {z 2 , z 3 } ∩ {z 5 , z 6 } = z k .Since z k can appear in the multi-set Z at most twice, it follows that c 2 has at least four neighbors different from z k and receives the charge by R1 and R2 from them at least ten times through the incident faces.Indeed, there are fourteen faces incident with {u, v} and at most four of them may be incident with z k .Since each vertex in Z − z k this time has degree at least 18 − d(z k ) 9, it follows that µ (c 2 ) = 5 − 6 + 5 − 6 + 10 × 1 3 > 0, as desired.

Lemma 2 .
No separating 3-cycle in G consists of three 5 − -vertices or has two 4-vertices.