Combinatorics and graph theory II.
3rd and 4th practice 29th of September and 6th of October, 2021 Planar graphs, Duality
Good to know:The dual of a planar graph G, which is denoted by G∗: Each face of G correspond to a vertex ofG∗and we put as many edges between two vertices ofG∗as many edges separate the two corresponding faces ofG. We draw these edges in such a way that each of them intersect one edge ofGand different edges of G∗ intersect different edges ofG.
H is analgebric dual ofGif there is a bijection between E(G) andE(H) which maps a cut to a cycle and a cycle to a cut.
H and Gareweakly isomorphic: if there is a bijection betweenE(G) and E(H) which maps a cut to a cut and a cycle to a cycle.
Whitney 1:Ghas an algebric dual if and only ifGis planar.
Whitney 2: LetGbe a planar graph and let H andGbe weakly isomorphic. Then:
1.H is planar
2.G∗ andH∗are weakly isomorphic, 3.GandG∗∗ are weakly isomorphic.
Whitney 3: Assume, thatGandH are weakly isomorphic. ThenH can be obtained fromGby the iterative application of the following three operations:
(a) Ifv is a cut vertex of the graph, then we cut the graph by the deletion ofv and put two seperate copies ofv back to the graph forming two connected components, one to each component.
(b) Two connected components are glued at a vertex.
(c) If the graph contains two vertices{u, v}which form a cutset, then we cut the graph among{u, v}, invert one connected component and glue the components among {u, v}: Lets fix a connected component C obtained by the deletion of{u, v}, so it was not a connected component before. Ifw∈V(C) andwwas adjacent tou(v) but it was not adjacent tov (u), then now it is adjacent tov (u) but it is not adjacent tou(v).
1. Are these graphs weakly isomorphic?
2. Prove that two trees are weakly isomorphic if and only if they have the same amount of vertices.
3. LetG(V, E) be a simple planar graph. Show thatEcan be partitioned toE1andE2such that (V, E1) and (V, E2) are bipartite graphs.
4. Let graphGandG∗ be finite simple graphs. We know thatGandG∗ are duals of each other. Show that, min{δ(G), δ(G∗)}= 3, whereδis the minimum degree.
5. LetGbe ann≥3 vertex simple plane graph which has 3n−6 edges. What is the maximum degree of the dual ofG?
6. LetFn =Kn,n−nK2 be the bipartite graph which we can obtain from Kn,n by deleting the edges of a perfect matching. For whichnisFn planar?
7. Assume that Gis a plane graph, each face ofG is a triangle and each face ofG∗ is a quadrilateral. How many edges and how many vertices doesGhave?
8. How big can be the chromatic number of a perfect planar graph?
9. Prove that for all simple planar graphs, containing at least three vertices, have at least three vertices whose degree is less than 6.
10. Gis a connected plane graph which has 200 edges. The dual ofGis a simple bipartite graph. Prove that Gcontains at most 100 vertices.
11. Gis a simple, connected plane graph.Ghasn≥3 vertices andGdoes not contain a cycle whose length is at most 5. Show thatG∗, the dual ofG, is not a simple graph.
12. LetGbe a connected planar graph. Give a plane graphG0 which is the dual of itself, soG0∗∼=G0 andGis a spanning subgraph ofG0.
13. LetGbe a plane graph and denote the number of faces ofGbyf =f(G). LetF1, F2, . . . , Ff be the faces of G, including the infinite face. Let|Fi| be the number of edges which boundFi where we count an edge twice if it separatesFifrom itself. Determine the maximum of
s(G) =
f
X
i=1
(|Fi| −1) ifGis a 10 vertex plane graph.
14. A connected graphG has 200 vertices and 300 edges. The dual of Gis simple. Show that the maximum degree ofGis 3.
15. If G is a plane graph, then let n(G), e(G) and f(G) be the number of vertices, edges and faces of G, respectively. Determine the maximum ofe(G)−n(G)−3t(G) whereGcan be any plane graph.
16. Let G be a plane graph. Prove that the faces of G can be colored by 2 colors (s.t. adjacent faces have different colors) if and only if all degrees ofGare even.
Homework
1. LetGbe a bipartite plane graph.G0 is created in the following way: Place a vertex at each face of Gand put an edge between two newly added vertices if their faces are adjacent. Furthermore we put an edge between each newly added verex and each vertex of the face where the newly added vertex has been placed. Prove that χ(G0)≤6.
Give a bipartite plane multigraphG such thatχ(G0) = 5 where G0 is obtained from Gby the construction which is given above.
2. LetGbe a plane graph and denote its degree sequence by d1, d2, . . . ,dn. Denote the number of faces of Gbyf =f(G). LetF1, F2, . . . , Ff be the faces of G, including the infinite face. Let|Fi|be the number of edges which boundFi where we count an edge twice if it separatesFi from itself. Let
s(G) =
f
X
i=1
(|Fi|+a) +
n
X
i=1
(di+a).
We know that the value ofs(G) is the same for each simple connected plane graphGif it contains at least three vertices. Determine the value ofa.
