By Xu B.G.

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**Additional resources for A 3-color Theorem on Plane Graphs without 5-circuits**

**Sample text**

David Wells, a British mathematician who is the author of many books on mathematics, puzzles, games and mathematics education, has displayed concern over the way mathematics is often taught to highschool students as well as to beginning college students. What Wells saw was the apparent goal of packing as much material as possible into each course while possibly neglecting aspects of mathematics that he (and many others) felt were important. At a number of universities, advanced mathematics students are able to participate in seminars, where they can see what the research faculty is studying.

14. Petersen’s drawing of the Petersen graph. ............... .... ... .... . . .... .......... .... . .... . . ...... .. . ......... .. . . . .......... . . .. . . . ... .. .. .... . . .. .. ... .. .............. .. . . .. ... .. .... .. ... ..... ... .. . . ....... .. .... ... ..... ..... .. ... ............... . . .... .. . ... ...........................................

22. 6 to find a 2-regular graph containing G as an induced subgraph. Find a smaller graph that has this property. 22 as an induced subgraph? 22a containing G as an induced subgraph. 22b containing G as an induced subgraph. Thus the minimum order of a graph with this property is 4. (b) Since G contains the end-vertices u and w, at least two vertices must be added to G to raise the degrees of u and w from 1 to 3. Because there is no 3-regular graph of order 5, it is necessary to add at least three vertices to G.