run multiple model and save model comparison results in dataframe in r

I want to run lm models and save model comparison result and extract p-values. I would like to save all the info in a dataframe. Using diamonds dataset as an example:

    diamonds %>% 
    group_by(cut) %>% 
    do(model1 = lm(price~carat, data=.),
       model2 = lm(price~carat+depth, data=.)) %>% 
    mutate(anova = anova(model2,model1)) %>% 
    mutate(pval= anova$'Pr(>F'[2])

I got error message below:

Error in mutate_impl(.data, dots) : 
  Column `anova` must be length 1 (the group size), not 6

My question is:

1. Why I got the error message and how to save anova result in the dataframe?
2. how to make the whole process work if lm or anova do not work on some subsets? something like try..catch..

My real data is more complicated then this. Just use diamonds and linear model to illustrate the idea. Thanks a lot.