Bipartite, -Colorable and -Colored Graphs
نویسنده
چکیده
A labeled graph is bipartite if its vertex set can be partitioned into two disjoint subsets and , = ∪, such that every edge of is of the form ( ), where ∈ and ∈ . Let be a positive integer and = {1 2 }. A labeled graph is colorable if there exists a function → with the property that adjacent vertices must be colored differently. Clearly is bipartite if and only if is 2-colorable. Define to be the number of -colorable graphs with vertices. We have 1 = 1 for ≥ 1 since a 1-colorable graph cannot possess any edges. We also have 1 = 1 for ≥ 1, 2 = 2 for ≥ 2, 32 = 7 by Figure 1, 33 = 8, 42 = 41 by Figure 2, and 43 = 63. More generally, −1 = 2(−1)2− 1 since the total number of labeled graphs with vertices is 2(−1)2 and, of these, only the complete graph cannot be (− 1)-colored. Does there exist a formula for ? The answer is yes if = 2, but evidently no for ≥ 3. We’ll examine this issue momentarily, but first define a related notion. A -colored graph is a labeled -colorable graph together with its coloring function. Let be the number of -colored graphs with vertices. The point is that a -colorable graph counts several times as a -colored graph. Clearly 1 = 1, 1 = , 22 = 6 by Figure 3, 23 = 15 by Figure 4, and 32 = 26 by Figure 5. When = 2, the following formulas can be proved [1, 2, 3]:
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تاریخ انتشار 2015