论文标题
Eulers Graph World-更多关于优雅边界的猜想-II
Eulers Graph World -- More Conjectures On Gracefulness Boundaries-II
论文作者
论文摘要
在本系列的第1部分中，研究了仅在（mod 4）操作下的欧拉图的子类。已经确定在规律性下的此类图对度> 2的程度不存在。在这里，我们考虑了（MOD 4）操作下只有两种循环的Euler图的子类。六个案件出现。当循环类型为0和2（mod 4）时，众所周知的二分欧拉图类别时，常规的两部分欧拉图的存在是众所周知的。在其他五种情况下，它会导致只有两种循环的常规欧拉图不存在。给出了带有该属性的Euler图的一些构造。我们研究了循环分解，块结构和循环交叉点的某些特性。 Euler图的循环分解允许隔离满足Rosa-Golumb标准的Euler图，因此不合格。在另一种情况下，图是可能的优雅候选人，并被猜想优雅，从而更好地理解了优雅的界限。循环类型为1＆2、1＆3、2和3（mod 4）的情况被证明是平面，在其他三种情况下，图形可能不是平面，其中包含非平面图的示例。对Euler图的属性进行探测是提出了预期的优雅边界，这些边界可能指导在哪里寻找优美的图形。此外，这些猜想可能会导致建立优美特性的分析技术。
The subclass of Euler graphs with only one type of cycles under (mod 4) operation was studied in Part-1 of this series. It was established that such graphs under regularity are nonexistent for degree >2. Here we consider the subclass of Euler graphs with only two types of cycles under (mod 4) operation. Six cases arise. The case when the cycle types are 0&2(mod 4), the well known class of bipartite Euler graphs, the existence of regular bipartite Euler graphs is very much known. In the other five cases, it transpires that regular Euler graphs with only two types of cycles are nonexistent. Some constructions of Euler graphs with the property are given. We investigate some properties of cycle decompositions, block structure and cycle intersections. Cycle decomposition of Euler graphs allows segregating Euler graphs satisfying Rosa-Golumb criterion and so are nongraceful. In the other case the graphs are possible candidates for gracefulness and are conjectured graceful leading to better understanding of gracefulness boundaries. The cases when the cycle types are 1&2, 1&3, 2&3(mod 4) the graphs are proved to be planar and in other three cases the graphs may not be planar with examples of nonplanar graphs. Probe into the properties of Euler graphs is to propose expected gracefulness boundaries which may guide where to look for graceful graphs. Further, the conjectures may lead to analytical techniques for establishing gracefulness property.
