Can my blog still be indexed by google without submitting a sitemap?

Since I just have a blog with tumblr, I can’t verify it on google sitemaps. How else can I get indexed so my posts appear on google serps? Should I just do like 10 backlinks on some do follow blogs?

I think you can verify your website in Google Webmaster which is more than enough and you can also generate site map in tumble. Just follow the below links which might be useful

Google Webmaster verification.

http://www.robinmalau.com/how-to-submit-your-tumblr-blog-to-google/

Sitemap

http://www.tothepc.com/archives/generate-xml-sitemap-of-tumblr-blog-submit-to-google/

General Guidelines

http://tumblelog.jauderho.com/post/122434303/4-tumblr-seo-tips

Don’t ignore link building which is very important

Good Luck:)