Tag Archives: enhanced power graph

The enhanced power graph is weakly perfect

Earlier this year, I posed a combinatorial problem, a solution to which would imply that, for any finite group G, the enhanced power graph of G is weakly perfect, that is, has clique number equal to chromatic number. Recall that … Continue reading

Posted in doing mathematics | Tagged , , , | 19 Comments

Why I’d like to see this solved

I am aware that quite a number of people have been captivated by the problem I posed. So here is the motivation for it, with some additional remarks and commennts. First, to repeat the problem: Problem: Let n be a … Continue reading

Posted in doing mathematics | Tagged , , | 3 Comments

Graphs on groups, 12

One thing I have learned from the project is that the most interesting question about graphs defined on groups is this: given two types of graph defined on a group G, what is the class of groups for which the … Continue reading

Posted in doing mathematics, exposition | Tagged , , , , | 1 Comment

Graphs on groups, 10

The lesson of this post and the next in the series is that the most interesting questions (to me, anyway) are not about the girth of the deep commuting graph but instead about the classes of groups G defined by … Continue reading

Posted in doing mathematics, exposition | Tagged , , , , , | Leave a comment

Graphs on groups, 4

Here is a small problem, mixing group theory and number theory, which might appeal to someone. A couple of definitions. The power graph of a group G has an edge from x to y if one is a power of … Continue reading

Posted in open problems | Tagged , , | Leave a comment

Graphs on groups, 2

I wrote the long post about this to try to write it out of my system. No luck … I mentioned in that survey that every finite graph is embeddable as induced subgraph in the enhanced power graph, deep commuting … Continue reading

Posted in doing mathematics, mathematics | Tagged , | Leave a comment