Weight of 3-paths in Sparse Plane Graphs

We prove precise upper bounds for the minimum weight of a path on three vertices in several natural classes of plane graphs with minimum degree 2 and girth g from 5 to 7. In particular, we disprove a conjecture by S. Jendrol' and M. Maceková concerning the case g = 5 and prove the tightness of their upper bound for g = 5 when no vertex is adjacent to more than one vertex of degree 2. For g 8, the upper bound recently found by Jendrol' and Maceková is tight.


Introduction
A normal plane map (NPM) is a plane pseudograph in which loops and multiple edges are allowed, but the degree of each vertex and face is at least three.
The degree of a vertex v or a face f , that is, the number of edges incident with v or f (loops and cut-edges are counted twice), is denoted by d(v) or d(f ), respectively.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 + -vertex v satisfies d(v) k, etc.
Let δ(G) be the minimum vertex degree, w k (G) be the minimum degree-sum of a path on k vertices (hereafter called a k-path) in a plane graph, and g(G) be its girth, that is the length of a shortest cycle.We will often drop the argument when the graph is clear from context.
An edge uv is an (i, j)-edge if d(u) i and d(v) j.More generally, a path v i . . .v k is a path of type (i 1 , . . ., i k ) if d(v j ) i j whenever 1 j k.
Already in 1904, Wernicke [20] proved that every NPM M 5 with δ(M 5 ) = 5 satisfies w 2 11, and Franklin [12] strengthened this to the existence of at least two 6 − -neighbors for a 5-vertex, which implies that M 5 satisfies w 3 (M 5 ) 17. Franklin's bound 17 is precise, as shown by putting a vertex inside each face of the dodecahedron and joining it with the five boundary vertices.
It follows from Lebesgue's results in [18] that each NPM has an edge of weight at most 14 incident with a 3-vertex, or an edge of weight at most 11, where 11 is sharp.For 3-connected plane graphs, Kotzig [17] proved a precise result: w 2 13.
Note that δ(K 2,t ) = 2 and w 2 (K 2,t ) = t + 2, so w 2 is unbounded if δ 2. Anyway, its finiteness may be enforced by certain additional constraints based, for example, on degree properties of particular subgraphs.For example, an induced cycle v 1 . . .v 2k in a graph is 2alternating (Borodin [3] This notion, along with its more sophisticated analogues (t-alternating subgraph, 3-alternator (Borodin, Kostochka, and Woodall [8]), cycle consisting of 3-paths (Borodin-Ivanova [9]), etc.), turns out to be useful for the study of graph coloring, since it sometimes provides crucial reducible configurations in coloring and partition problems (more often, on sparse plane graphs, see Borodin [10]).Its first application was to show that the total chromatic number of planar graphs with maximum degree ∆ at least 14 equals ∆ + 1 (Borodin [3]).
Nowadays, the maximum weight of edges is known for all most interesting classes of plane graphs with given girth (further examples and references can be found in Borodin [4,5,6,10]).
We now switch to the maximum weight w 3 of 3-paths.In 1993, Ando, Iwasaki, and Kaneko [1] proved that every 3-polytope satisfies w 3 21, which is sharp due to the Jendrol' construction [15].Jendrol' [14] proves that each 3-polytope has a 3-path uvw such that max{d(u), d(v), d(w)} 15 (the bound is precise).Jendrol' [15] further shows that such a path must belong to one of ten types, in which d(u) + d(v) + d(w) varies from 23 to 16.
Note that the graphs of 3-polytopes are precisely the 3-connected planar graphs due to Steinitz's famous theorem [19].The requirement of 3-connectedness is essential for the finiteness of w 3 , as shown by the construction K * 2,2t obtained from the double 2t-pyramid by deleting a t-matching from the 2t-cycle formed by 4-vertices (Borodin [7]).
Moreover, Borodin [7] showed that only the presence in a NPM of K * 2,4 is responsible for the unboundedness of w 3 .The following refinement of the bound w 3 21 by Ando, Iwasaki, and Kaneko [1] holds: Theorem 1 (Borodin [7]).Every normal plane map without K * 2,4 has (i) either w 3 18 or a vertex of degree 15 adjacent to two 3-vertices, and (ii) either w 3 17 or w 2 7.
As mentioned above, the bounds w 3 21 and w 3 17 are tight.For a long time, it was not known whether the bound w 3 18 in Theorem 1 is sharp or not; its sharpness was recently confirmed in Borodin et al. [11].In particular, Ando, Iwasaki, and Kaneko's [1] precise bound w 3 21 is valid for all NPMs with w 2 > 6 (Borodin [7]).Also, Theorem 1 immediately implies that Franklin's precise bound w 3 17 is valid for all normal plane maps with δ 4.
Recently, Borodin et al. [11] precisely described 3-paths in all normal plane maps without K * 2,4 (in particular, in planar graphs with δ 3 and in 3-polytopes) by showing that they belong to eight specific types having weight from 17 to 21, where all parameters are best possible.
Note that the star graph K 1,n satisfies δ = 1 and w 3 = n + 2. The behavior of 3-paths with low degree-sum in sparse planar graphs with δ = 2 was recently studied by Jendrol' and Maceková [16].As observed in [16], if we join vertices a and b by independent paths ax i y i b with 1 i n, then w 3 = n + 4.

Theorem 2 ([16]
).Every planar graph G with δ = 2 and girth g(G) g 5 has a 3-path of one of the following types: Also, they conjectured that the bound in Corollary 3(i) can be lowered to 9.
The purpose of this paper is to establish precise upper bounds on w 3 whenever 5 g 7 under the assumptions of Corollary 3, and also in a broader class of planar graphs with g = 6.In particular, we disprove Conjecture 4.
Our new results are in Theorems 5-7 below.
We see that Theorem 6 extends Corollary 3(ii) and improves the upper bound in it.
Theorem 7. Every plane graph with δ = 2 and g 7 has w 3 9, which bound is tight whenever 7 g 9.
So, Theorem 7 improves the upper bound in Corollary 3(iii).
More specifically, the bounding cycle of the graph to be obtained may be encoded as 5, 3, 5, 3, . . .according to the degrees of its vertices.Moreover, its internal half may be encoded as 5 2 , 3 1 , 5 1 , 3 0 . .., where the subscripts show the number of ingoing edges.For the exterior half, we have a similar encoding 5 1 , 3 0 , 5 2 , 3 1 . .., so the two halves can be glued in this order.

Proof of Theorems 6 and 7
Proof of Theorems 6 and 7. Forbidding (2, 2, ∞, 2)-paths in Theorem 6 is justified by the already mentioned graph with w 3 = ∞ and g = 6 in which vertices a and b are joined by independent paths ax i y i b with 1 i n.We note that forbidding (2, 2, ∞, 2)-paths still allows both (2, 2, ∞)-paths and (2, ∞, 2)-paths.The sharpness of the bound on w 3 follows by putting a 2-vertex on every edge of the icosahedron, which results in w 3 = 2 + 5 + 2 under the absence of (2, 2, ∞, 2)-paths.
To confirm the tightness of Theorem 7, we put two 2-vertices on every edge of the icosahedron.If desired, we can then fix one of 9-faces and contract any two 2-vertices in its boundary to obtain g = 7.

Discharging and its consequences
Let M be a counterexample to the upper bounds on w 3 in Theorems 6 or 7. Without loss of generality, we can assume that M is connected.Let V , E, and F be the sets of vertices, edges and faces of M , respectively.Euler's formula |V | − |E| + |F | = 2 for M may be rewritten as Every vertex v contributes the charge µ(v) = d(v)−6 to (1), so only the charges of 5 −vertices are negative.Every face f contributes the non-negative charge µ(f ) = 2d(f ) − 6 to (1).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 (x) is non-negative for all x ∈ V ∪F .This will contradict the fact that the sum of the new charges is, by (1), equal to −12.
Throughout the paper, we denote the vertices adjacent to a vertex or incident with face x in a cyclic order by v 1 , . . ., v d(x) .Let ∂(f ) be the boundary of a face f , n i (f ) be the number of i-vertices in ∂(f ), and ρ(f 2 n 4 (f ).Now we apply the following rule of discharging.

R. Every face f gives to each incident vertex
To complete Case 1, we observe that any other 6-face is either a (2, 2, 4 + . ..)-face or (2 + , 3 + , 3 + . ..)-face.In both cases, f gives at most 9  2 to its three vertices of smallest degrees, which means that each incident 5-vertex receives 1  2 from f , as stated in Lemma 8(iii).
It remains to assume that 4 d(v 3 ) 5 (see Fig. 3d), for otherwise f 2 already gives

Figure 3 :
Figure 3: Every 5-vertex receiving 0 from an incident 6-face receives at least 1 in total.