Hello,


Thank you for taking a look at this post. I am trying to prove the statement as given in the title. I have managed to disprove it but I am yet not sure if it is correct. I would appreciate it if somebody could comment if the above statement is true or false. Thank you for your time.



For two graphs, $H$ and $G$ let $\bar{H}$ and $\bar{G}$ be their respective compliments. Then if $H$ is a subgraph of $G$, then $\bar{H}$ is a subgraph of $\bar{G}$.
This statement is not true. For example, let $G=K\_4$ and $H=C\_4$, where $G$ and $H$ are complete graph and cycle graph with four vertices, respectively.
It is easy to check that, $H$ is a subgraph of $G$, but $\overline{H}$ is not a subgraph of $\overline{G}$.


It is interesting that, you think about this question:


When this statement is true? 


