Havel-Hakimi theorem

<aside> ❗ Let $[s, t_1, t_2,...,t_s, d_1, d_2,...,d_n]$ be an ordered sequence. This sequence is graphic $\Leftrightarrow$ the sequence $[t_1-1, t_2-2,...,t_s-1, d_1, d_2, ...,d_n]$ is graphic.

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Algorithm (given a sequence $A$)

  1. Repeatedly reduce the size of $A$ by applying the “only if direction” $\Rightarrow$ if the theorem untill we reach to a small-size sequence $B$
    1. If $B$ is graphic → $A$ is graphic
    2. If $B$ is not graphic → $A$ is not graphic

Application

Show that $[4, 4, 2, 2, 2]$ is graphic

Untitled

Proof

The theorem can be proved using proof by cases. In which we first prove the “if part” $\Leftarrow$ and then the “only if part” $\Rightarrow$.

Walks, Paths and Cycles

Walks

Given a graph $G$ and two vertices $v_0, v_k\in V(G)$ a $(v_0, v_k)$-walk in $G$ is a sequence $[v_0, e_1, v_2, e_2, ...,e_k, v_k]$ of vertices and edges such that $e_i=[v_i-1, v_i]$

Plain English Finite or infinite sequence of edges which joins a sequence of vertices

Trail

Walk in which all edges are distinct

Path

Trail in which all vertices (and therefore also all edges) are distinct

Cycle

A cycle is a non-empty trail in which only the first and last vertices are equal. Note - A self loop is a cycle.

Simple graphs

For simple graph a cycle is a $(u,u)$-path of at least 3 edges.

Connectivity