# Download PDF by Xu B.G.: A 3-color Theorem on Plane Graphs without 5-circuits

By Xu B.G.

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3. 6 Matrices Associated with a Graph 33 Fig. 29 A graph with oriented members and cycles 10 3 2 C3 9 C2 7 6 8 1 C1 5 4 Theorem. Let S have an incidence matrix B and a cycle basis incidence matrix C. Then: CBt ¼ 0ðmod 2Þ: ð1:60Þ A simple proof of this theorem can be found in Kaveh [10]. Notice that Eq. 60 holds due to the orthogonality property discussed in Sect. 3. In fact, the above relation holds even if the cutsets or cycles do not form bases, or the matrices contain additional cutsets and/or cycle vectors.

If both are connected, the cutset is called prime. If one of S1 or S2 consists of a single node, the cutset is called a cocycle. These definitions are illustrated in Fig. 20. 7 23 Trees, Spanning Trees and Shortest Route Trees A tree T of S is a connected subgraph of S which contains no cycle. A set of trees of S forms a forest. Obviously a forest with k trees contains N(S) À k members. If a tree contains all the nodes of S, it is called a spanning tree of S. Henceforth, for simplicity it will be referred to as a tree.

When the directions of the cycles are taken as those of their corresponding chords (dashed lines), the fundamental cycle basis incidence matrix can be written as:  2 3 C1 1 À1 0 0  1 0 0 ð1:68Þ C ¼ C2 4 1 0 1 0  0 1 0 5: C3 1 À1 1 À1  0 0 1 It should be noted that the tree members are numbered first, followed by the chords of the cycles in the same sequence as their generation. Obviously, BCt ¼ CBt ¼ 0ðmod 2Þ, with a proof similar to that of the non-oriented case. A cuset basis incidence matrix is similarly obtained as:  2 3 1 0 0 0  À1 1 À1 60 1 0 0 1 0 1 7  7 CÃ ¼ 6 4 0 0 1 0  0 1 À1 5,  0 0 0 1 0 0 1 CÃT CÃc ð1:69Þ ð1:70Þ where the direction of a cutset is taken as the orientation of its generator (the corresponding tree member).