on the effect of major vertices on the number of light edges

On the Effect of Major Vertices on the Number of Light Edges

Daniel P. Sanderst DEPARTMENT OF MATHEMATICS

THE OHIO STATE UNIVERSITY COLUMBUS, OHIO 43210

ABSTRACT

This paper presents an inequality satisfied by planar graphs of minimum degree five. For the purposes of this paper, an edge of a graph is light if the weight of the edge, or the sum of the degrees of the vertices incident with it, is at most eleven. The inequality presented shows that planar graphs of minimum degree five have a large number of light edges. This inequality improves upon a recent inequality of Borodin and Sanders, which showed that 7/15 times the number of edges of weight 10 plus 1/5 times the number of edges of weight 11 is at least 12. These constants 7/15 and 1/5 were shown to be best possible. The inequality in this paper shows that, for this type of graph, the presence of vertices of degree at least eight increases the number of light edges. A graph is presented which shows that the coefficient obtained for the number of degree eight vertices is best possible. 0 1996 John Wiley & Sons, Inc.

1. INTRODUCTION

This paper deals only with plane graphs which have minimum degree five and which have no face with at most two edges in its boundary.

Let an i-vertex be a vertex of degree i. Let an i,j-edge be an edge joining an i-vertex to a j-vertex. Given a plane graph G, let ui be the number of i-vertices of G, and let ei, be the number of i,j-edges of G. Improving upon earlier results of Wernicke [5], Grunbaum [3], Fisk [4], Grunbaum and Shephard [4], and Borodin [ 1 1 , an inequality was proven for planar graphs of minimum degree five by Borodin and Sanders [2]:

+This research was supported by the Office of Naval Research, Grant NOOO14-92-J-1965.

Journal of Graph Theory, Vol. 21, No. 3, 317-322 (1996) 0 1996 John Wiley & Sons, Inc. CCC 0364-9024/96/030317-06

318 JOURNAL OF GRAPH THEORY

This inequality was also shown to be best possible in the sense that the coefficients of e5,5

and e5.6 cannot be replaced by smaller constants while keeping the inequality true. This was done by the demonstration of two graphs, one having e5.5 = 0 and e5.6 = 60, and the other having e5.5 = 24 and e5.6 = 4. The existence of the light edge inequalities of this nature was originally discussed in relation to the Four Color Problem, and later found useful with regards to acyclic colorings.

A simple manipulation of Euler’s formula shows that u5 5 12 for planar graphs of minimum degree five. The number of vertices of degree five is affected by the number of vertices in the graph with large degree. In fact, the following inequality is true:

The number of vertices of large degree has a surprisingly similar effect to the number of light edges in a planar graph of minimum degree five. Section 3 proves the following inequality:

(7/15)e5,5 + (1/5)e5,6 I 12 + (23/30)u~ f 2kE9((5/6)k - 6)vk.

The coefficient on U g is shown to be best possible. The theorem is proven by a discharging method, the rules of which are described in Section 2. The discharging rules and the proof of the inequality are a direct extension of the discharging rules and the proof of the inequality of Borodin and Sanders [2].

2. DISCHARGING

Let a plane graph G be charged if for every vertex x of G , x is assigned a charge equal to 6 - deg(x). For a charged plane triangulation G , note that Euler’s formula implies that the sum of the charges of the vertices of G is twelve. Let a vertex of a plane graph with charges be overcharged if it has a positive charge; clearly the overcharged vertices of a charged plane graph are precisely its vertices of degree at most five. Also, let a major vertex be a vertex of degree at least seven, while a minor vertex has degree at most six.

Given a plane graph G , and a vertex x of G, let a cyclic neighborhood of x be a list of its neighbors according to one of the two cyclic permutations defined by the planar embedding. Similarly, let a degree sequence of x be a list of the degrees of its neighbors according to one of the two cyclic permutations defined by the planar embedding. Let a 7-vertex x of a plane triangulation be surrounded if x has degree sequence 5,5,5,5,5,5,5. Let a 7-vertex be saturated if it is surrounded, and six of its neighbors have degree sequence 5,7,5,7,5. Let a 7-vertex be semi-saturated if it is surrounded, four of its neighbors have degree sequence 5,7,5,7,5, two of its neighbors have degree sequence 5,7,5,8,5, and the last neighbor has degree sequence 5,7,5 ,I ,7.

Let a plane graph of minimum degree five be discharged if the graph is first charged, and then the charges are changed as follows: Add a charge of 23/30 to each vertex of degree 8. For each k 2 9, add a charge of (5/6)k - 6 to each vertex of degree k . Then the charges on the overcharged vertices (5-vertices) are redistributed as follows: Let u be a vertex of degree five. Send a charge of 49/210 to each 5,5-edge incident with u. Send a charge of 42/210 to each 5,6-edge incident with u. The remaining positive charge at u will be sent to the major vertices surrounding u as follows:

MAJOR VERTICES AND LIGHT EDGES 319

For each major neighbor x of u, and for each vertex y of degree at least six adjacent to both u and x , send a charge of 15/210 to x via y . For each neighbor z of u such that deg(z) 2 9, send 35/210 to z . If u has degree sequence 5,7,5,8,5, and the 7-vertex is surrounded, send 29.75/210 to the 7-vertex and 33.25/210 to the %vertex. If u has degree sequence 7,5,7,6,5, send 28/210 to the first 7-vertex and 27/210 to the second 7-vertex. If u has degree sequence 7,5,7,8,5, send an extra 0.75/210 to the %vertex via the second 7-vertex, send 31.5/210 directly to the %vertex, send 22.75/210 to the first 7-vertex, and send 27/210 to the second 7-vertex. If u has degree sequence 7,5,8,8,5, send 31.5/210 to each 8-vertex, and send 19/210 to the 7-vertex. If u has degree sequence 5,8,5,7,7, and the %vertex is adjacent to at least two vertices of degree at least six, then send 35/210 to the &vertex, and 23.5/210 to each of the 7-vertices.

Let u have degree sequence 7,5,7,7,5, and let s be the first 7-vertex. If s is adjacent to at least two vertices of degree at least six, then send 35/210 to s, and 23.5/210 to each of the other 7-vertices. If s is semi-saturated, then send 24.5/210 to s. Let s1 and s2 be the other 7- vertices, such that there is an %vertex distance two from both s and s1. If s1 has two neighbors with degree sequence 5,7,5,7,5, then send 27/210 to s1 and 30.5/210 to 82. Otherwise send 30.5/210 to s1 and 27/210 to s2. If s is saturated, then send 21/210 to s. Let f1 and t 2 be the other 7-vertices. If tl is adjacent to six 5-vertices, while 12 is not, then send 271210 to 11 and 34/210 to 12. Symmetrically, if 12 is adjacent to six 5-vertices, while 11 is not, then send 34/210 to t l and 27/210 to t 2 . In any other case, send 30.5/210 to each of t l , 22. If none of the previous cases occurs, send 28/210 to s, and 27/210 to each of the other two 7-vertices.

Finally, if v is in none of the situations previously mentioned, split the remaining positive charge at u evenly among its neighbors of degree 7 or 8.

3. LIGHT EDGES AND MAJOR VERTICES

This section uses the discharging rules of Section 2 to prove the main result of this paper, the inequality mentioned in Section 1.

Theorem. Every planar graph of minimum degree five satisfies the following:

Furthermore, the coefficients- on e5.5, e5.6 and us are each best possible. Proof. Let G be a plane graph of minimum degree five. Without loss of generality (by

adding edges), G is a triangulation. Let G be discharged. First notice, from the definition of discharged, that no minor vertex is overcharged. Each vertex of degree at least nine receives at most 35/210 from each of its neighbors, and is thus not overcharged. Thus if there is an overcharged vertex, it must have degree 7 or 8.

Assume that there is an overcharged %vertex x . Before receiving charge from the 5-vertices, an %vertex has charge -259/210. Thus if x is overcharged, at least one of its neighbors must send in more than 32.375/210. One case is if x has at least two neighbors y and z which are not 5-vertices. Here, some vertices could send in 35/210 to x . But note that each of y , z sends in at most 31.5/210 to x . If y is adjacent to z , then each of y , z sends in at most 15.75/210, so x would not be overcharged. If there is a vertex w adjacent to all of x , y , z then w sends in at most 20/210. If a, b are the other neighbors of x adjacent to one of y , z , then note that a , b each send in at most 31.5/210, and x cannot be overcharged. Finally, let c , d , e , f be the distinct neighbors of x adjacent to one of y , z ; here each of c , d , e , f sends in at most 31.5/210, and x again cannot be overcharged.

320 JOURNAL OF GRAPH THEORY

Now every neighbor of x sends in at most 31.5/210 unless it is a 5-vertex with degree sequence 5,7,5,8,5, where the 7-vertex is surrounded. This type of vertex sends in 33.25/210. For the total of the charge going in to be higher than 259/210, x must have at least five 5-vertices of this type as neighbors. In order for this to occur, there must be at least three surrounded 7-vertices at distance two from x . The only way this can happen, by planarity, is with a structure symmetric to the following. Let U I , u2, u3, u4, u5, U6, u7, U S , be the cyclic neighborhood of x . One of the three surrounded 7-vertices is adjacent to both u1 and u2;

another is adjacent to both u3 and u4; the last is adjacent to both V6 and u7. For there to be at least five Svertices, sending in 33.25/210, both of u5 and are 5-vertices with degree sequence 5,5,5,5,8. But then, each sends in 14/210 to x , and x is not overcharged.

Assume that there is an overcharged 7-vertex x . For x to be overcharged, some neighbor must send in more than 30/210. From the definition of discharged, there are only a few ways that this can occur. These will be considered one at a time in the following.

Suppose x has a neighbor z which is a 5-vertex with degree sequence 7,5,7,7,5, where the first 7-vertex t is saturated or semi-saturated, and x is the second 7-vertex. Suppose z sends in either 30.5/210 or 34/210 to x . Let the cyclic neighborhood of x be y , z , U I , u2, u3, u4, w, where deg(y) = 7. Let the cyclic neighborhood of y be x , z , u1, u2, u3, u4, w. Since t is either saturated or semisaturated, each of z , U I , u2, u1, u2 is a 5-vertex. By symmetry, no neighbor of x , except possibly z , can send in 34/210 to x . Also note that u3 is the only vertex which could send in 35/210, and if it did, w and y would each send in at most 15/210, and x is not overcharged. From the definition of discharged, each of ul, u2, u3, u4 send in at most 31.5/210 to x , while w sends at most 30.5/210 and y sends at most 30/210. Since x is overcharged, this means that each of u1, u2, u3, u4, w sends in more than 20/210 to x .

If deg(w) > 5 , then w would send in at most 15/210 to x ; thus deg(w) = 5. If deg(u3) > 5 , then u2 would send in at most 20/210 to x ; thus deg(u3) = 5. If either u4 or u4 has degree greater than 5, then w would send in less than 20/210; thus deg(u4) = 5 and deg(u4) = 5. This shows that z sends in exactly 30.5/210 to x . Let the cyclic neighborhood of u4 be

Taking this into consideration, deg(u5) = 5, for if it was not, both of u3 and u4 would send in at most 28/210 to x , a contradiction.

Consider the case where t is saturated. By definition of discharged, y must also have six neighbors of degree five, and deg(u3) = 5. Finally, planarity gives that U6 has at least two neighbors which have degree at least six. This last result shows that w sends in 23.5/210 to x , and x is not overcharged.

Consider the case where t is semi-saturated. Let the cyclic neighborhood of u4 be w , y , u3, u5, U 6 . By definition of discharged, u3 and u5 are both 5-vertices. Planarity gives that U 6 is either of degree at least nine, or has at least two neighbors which have degree at least six. Again, this shows that w sends in 23.5/210 to x , and x is not overcharged.

Consider the case when a charge 35/210 is sent into x . In this case, x has at least two neighbors y and z , which are not 5-vertices. Note that each of y , z sends in at most 30/210 to x . If y is adjacent to z , then each of y , z sends in at most 15/210, so x would not be overcharged. If there is a vertex w adjacent to all of x , y , z , then w sends in at most 20/210. If a, b are the other neighbors of x adjacent to one of y, z, then note that a, b each send in at most 27/210, and x cannot be overcharged. Finally, let c, d , e , f be the distinct neighbors of x adjacent to one of y , z ; here each of c , d , e , f sends in at most 27/210, and x again cannot be overcharged.

The only remaining case where a neighbor sends in more than 30/210 is if 31.5/210 is being sent in. If x receives this amount from at least five of its neighbors, then it receives

w , X, V 3 , ~ 5 , U 6 . If deg(U6) 5 6, then w would send in at most 20/210 to X; thus deg(U6) > 6.

MAJOR VERTICES AND LIGHT EDGES 321

- FIGURE 1

it from six of its neighbors, and it is saturated. Consider instead the case where six of the seven neighbors send in either 31.5/210 or 29.75/210, as the structures are similar. If x is saturated, then the remaining neighbor sends in at most 21/210, and x is not overcharged. If x is semisaturated, then the remaining neighbor sends in at most 24.5/210, and x is not overcharged. If x has a neighbor with degree sequence 7,5,7,8,5, then it sends in 22.75/210 to x, and x is not overcharged. If x has a neighbor with degree sequence 7,5,8,8,5, then it sends in 19/210 to x, and x is not overcharged. In the final case here, four of the neighbors of x have degree sequence 5,7,5,8,5. In this case, the last neighbor sends in at most 28/210, and x is not overcharged.

If x receives either 31.5/210 or 29.75/210 from four of its neighbors, there are two cases: If x is adjacent to a vertex y of degree greater than five, then y sends in at most 30/210, and each of the two vertices adjacent to both x and y send in at most 27/210. If x is adjacent only to vertices of degree five, then the other three neighbors send in at most 28/210. In either case, x is not overcharged.

The last case is where x receives 3 1.5/210 or 29.75/210 from less than four of its neighbors. Note that each neighbor of x of degree five which does not send in one of these charges sends in at most 28/210 to x. If there are at least three of these, then x is clearly not overcharged. There must be at least two neighbors of x of degree five which do not send in 31.5/210 or 29.75/210. If that is all, then x is adjacent to two adjacent vertices of degree greater than five; each sends in at most 15/210. Thus there is no overcharged vertex.

A charged graph has the sum of the charges of its vertices equal to 12. A discharged graph has the sum of the charges of its vertices plus the charges of its edges at least r : = 12 + (23/30)u~ + &9((5/6)k - 6)uk. The previous argument showed that no vertex is overcharged, or that the sum of the charges of the vertices is at most 0. Thus, the sum of the charges of the edges is at least r . From the definition of discharged, the sum of the charges of the edges is equal to (7/15)e5,5 + (1/5)e5,6.

That the coefficient of e5.6 is best possible follows from a graph in [I] which has e5.5 = 0, e5.6 = 60, and for every k 2 8, has U k = 0. That the coefficient of e5.5 is best possible follows from a graph in [2] which has e5.5 = 24, e5.6 = 4, and for every k 2 8, has Uk = 0. That the coefficient of ug is best possible follows from the graph of Figure 1, which has e5.5 = 29, e5,6 = 0, us = 2, and for every k 2 9, has t f k = 0. 1

322 JOURNAL OF GRAPH THEORY

4. OPEN QUESTIONS

First note that the graph in [2] which was mentioned has V 6 > 0 and u7 > 0, and thus terms for u6 or u7 cannot be introduced into the inequality. Of the additional terms which were presented in the Theorem, only one, ug, has a coefficient which is known to be best possible. An obvious open question is to determine the best possible values for these coefficients. The Theorem gives a lower bound for the coefficients. The following Proposition establishes an upper bound for some of the coefficients. For k 2 9, let ck be the best possible coefficient for Vk.

Proposition. For k 2 9, if k is divisible by 3, then ck I (77/90)k - 6.

Proof: Let a k 2 9 which is divisible by 3 be given. Let the graph Gk be defined as follows: The graph has two vertices x and y of degree k, each adjacent only to vertices of degree five. Let the cyclic neighborhood of x be u1, u2, . . . , U k . For each i such that 1 5 i 5 k and i is divisible by 3, let ui have degree sequence 5,7,7,5,k. For each i such that 1 I i I k and i is not divisible by 3, let y i have degree sequence 5,5,7,5,k. The neighborhood of y is identical. Each vertex of degree 7 is adjacent to two neighbors of x and two neighbors of y. The resulting graph has u5 = (8/3)k, u7 = (2/3)k, and Vk = 2, and these are all vertices of Gk. This graph has e5.5 = (11/3)k and e5.6 = 0. The resulting inequality for this graph, with its zero terms omitted, is: (7/15) (1 1/3)k 2 12 + 2Ck. Solving for Ck obtains the inequality in the statement of the Proposition. I

Upper bounds asymptotically, the same to that of the Proposition can be obtained for k not divisible by 3. For k divisible by 3, the following aesthetic bounds have now been obtained:

These constants appear to offer some insight into the structure of planar graphs of minimum degree five. The correct asymptotic value for large k especially seems interesting.

The inequality of the Theorem can be improved by adding extra terms which do not seem as interesting as the ones that were included. In particular, terms such as the following could be added to the right hand side of the inequality without changing the proof: 1/42 times the number of edges from a vertex of degree 6 through 8 to a vertex of degree at least 9, 1/21 times the number of edges between two vertices of degree at least 9, l/7 times the number of vertices of degree at least 9 in each triangle which has each vertex incident with it of degree at least 6. These terms were not included, because they would almost surely not appear in the inequality with the best possible coefficients (the cks). Also, for odd k, a constant of 1/30 could be added to the coefficient in the inequality without changing the proof much, but this seemed awkward.

References

[ l ] 0. V. Borodin, Structural properties of planar maps with the minimal degree 5, Math

[2] 0. V. Borodin and D. P. Sanders, On light edges and triangles in planar graphs of

(31 B. Grunbaum, Acyclic colorings of planar graphs, Israel J. Math. 14 (1973), 390-408. [4] B. Grunbaum and G. C. Shephard, Analogues for tilings of Kotzig’s theorem on minimal

[5] P. Wernicke, h e r den kartographischen Vierfarbensatz, Math. Ann. 58 (1904), 413-426.

Nuchr. 158 (1992), 109- 117.

minimum degree five, Math. Nachr. 170 (1994), 19-24.

weights of edges, Ann. Discrete Math. 12 (1982), 129-140.

Received August 2, 1994