# question about simple graph.

I know that simple graph has no parellel edges and loops. My question is that I have to draw the graph on six vertices with degree sequence $(3,3,5,5,5,5)$. I draw the the graph with the given degree sequence and everytime I got the graph which is not simple. Does there exist a simple graph with these degree sequence? If no why?

-

Do you know the Havel-Hakimi Theorem? – EuYu Sep 18 '12 at 18:14
No! Can you please state this theorem! – Kns Sep 18 '12 at 18:17

## 2 Answers

No fancy theoretical machinery is required.

Call the vertices $v_1,v_2,v_3,v_4,v_5$, and $v_6$. Let $v_1,v_2,v_3$, and $v_4$ be the vertices of degree $5$. There are only six vertices, so each of these vertices must be connected by an edge to every other vertex in the graph. For instance, $v_2$ must be connected to $v_1,v_3,v_4,v_5$, and $v_6$. This means that $v_5$ and $v_6$ are both connected to each of the vertices $v_1,v_2,v_3$, and $v_4$ and must therefore have degree at least $4$: the degree sequence $(3,3,5,5,5,5)$ is impossible for a simple graph.

If you’d like to explore further, this kind of reasoning leads to the easy half of the Erdős-Gallai theorem, which gives a simple computational criterion for whether a sequence is the degree sequence of some simple graph. This kind of reasoning shows that a sequence that fails the criterion cannot possibly be a degree sequence; proving that every sequence that satisfies it really is the degree sequence of some simple graph is harder. See also the comments here.

-

By the Havel-Hakimi Theorem, a descending degree sequence $\{a_1, \cdots ,\ a_k\}$ is graphical if and only if $a_1 \le k-1$ and $\{a_2 - 1, \cdots, a_{a_1+1}-1, a_{a_1 +2}, \cdots, a_k\}$ is graphical.

Applying the theorem to your case, we have $$\{5,5,5,5,3,3\} \iff \{4,4,4,2,2\}\iff\{3,3,1,1\}\iff\{2,0,0\}$$ The latter graph is evidently impossible.

-

Evidently not. $\{1,1,1,1\}$ is the degree sequence of two disjoint $K_2$s. However, it is impossible for the sequence as originally stated. – Rick Decker Sep 18 '12 at 19:46
@RickDecker Sorry about that, I must've mistyped the sequence in the middle and started using it. It is fixed now. – EuYu Sep 18 '12 at 20:28
A minor point, but if you edit a post of yours it's a good idea to indicate the edit as part of your post. Otherwise someone new to this thread might be confused by the existing comments (as here, for example). – Rick Decker Sep 18 '12 at 23:38
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 20, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9546744227409363, "perplexity_flag": "head"}