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I'm reading about covering space ,i'm having the following two doubts:

1.Suppose $X$ be a graph and $\widetilde {X}$ be a regular cover/Normal Cover of $X$ other than $X$.Then show that $\widetilde{X}$ is a regular graph (i.e. each vertex has same degree).

2. Suppose if $X$ is finite then is it true that every covering graph of $X$ is regular graph?

I know that $\widetilde {X}$ is a graph but i can't see that its a regular graph.

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Math Lover
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  • Statement (1) is not true in general...for instance, any space is a normal covering of itself, so this would be saying any graph is regular... – Eric Wofsey Dec 15 '16 at 05:50
  • I see..do you have any other example in mind?(since i think its not an interesting example).I've edited the question.Thanks! – Math Lover Dec 15 '16 at 05:52

1 Answers1

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"Regular" in the covering sense and "regular" in the graph sense are basically totally unrelated. Statement (1) is horribly false. Indeed, if $p:\widetilde{X}\to X$ is a covering of graphs, then for any vertex $v\in\widetilde{X}$, $v$ has the same degree as $p(v)$ (this is basically immediate from the fact that $p$ is a local homeomorphism, since the degree of a vertex is a local topological property). So, if you require your covering maps to be surjective, $\widetilde{X}$ is a regular graph iff $X$ is a regular graph (since their corresponding vertices have the same degrees).

Eric Wofsey
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  • I see.. thank you.Btw,this question was motivated from the first answer here by user641.Could you please have a look?[ I think its just because $X$ is bouquet of circles?] – Math Lover Dec 15 '16 at 06:32
  • Yeah, the statement in that answer is very misleading. It seems the author was really using "the degree of each vertex is the same, etc." as an abbreviation for much stronger properties that are special to normal coverings, even though the only example property they named is actually true for all coverings (since in this case $X$ is a wedge of circles, which is a regular graph). – Eric Wofsey Dec 15 '16 at 06:47
  • But that is not a consequence of regularity right? Where does he uses regularity in solution? – Math Lover Dec 15 '16 at 06:50
  • His "solution" is extremely sketchy, and he hasn't actually explained how he's using regularity. – Eric Wofsey Dec 15 '16 at 06:58
  • Yeah right.So i think answer to my question 2 is also negative as any finite graph which is not regular would do the job? – Math Lover Dec 15 '16 at 07:01
  • That's correct. – Eric Wofsey Dec 15 '16 at 07:03
  • Thank you,i appreciate your help! – Math Lover Dec 15 '16 at 07:03