Eulers图世界--关于优美性边界的更多猜想-- III
Eulers Graph World -- More Conjectures On Gracefulness Boundaries-III
摘要：
第一部分（II）研究了在（mod 4）运算下只有一（2）类圈的欧拉图。在这里，我们考虑一类欧拉图只有三种类型的循环下（模4）。这导致了四种情况，即，循环类型为（0，1，2），（0，1，3），（0，2，3），（1，2，3）的图。给出了每类欧拉图的一些构造。在部分情况下证明了二度结点的存在性，并猜想p>5阶图的二度结点的存在性。作为一个特例，我们猜想，正则性是不存在的，在所有四种情况下，阶p>5的图。也就是说，不存在p>5阶的正则欧拉图恰好有三种圈。换句话说，一个p>5阶的正则欧拉图有三种类型的圈，它有一个第四种类型的圈。给出了前三种情况下非平面欧拉图的一个例子。满足Rosa-Golomb准则的这类图是非竞争图。在其他情况下，给出了图的优美性的必要条件，从而更好地理解了图的优美性边界。
英文摘要：
Euler graphs with only one (two) type(s) of cycles under (mod 4) operation were studied in Part-I(II). Here we consider the class of Euler graphs with only three types of cycles under (mod 4). This gives rise to four cases viz., graphs having cycle types (0,1,2), (0,1,3), (0,2,3), (1,2,3). Some constructions of Euler graphs of each class are given. We prove the existence of degree two node in part cases and conjecture the existence of degree two node in every graph of order p>5. As a special case, we conjecture that regularity is nonexistent in all four cases for graphs of order p>5. That is, regular Euler graphs of order p>5 with exactly three types of cycles donot exist. In other words, a regular Euler graph of order p>5 with three types of cycles has a cycle of fourth type. An example of a nonplanar Euler graph in the first three cases is given. Graphs of this type satisfying Rosa-Golomb criterion are nongraceful. In other cases necessary conditions are given and the graphs are conjectured graceful leading to better understanding of gracefulness boundaries.
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