Codex @codex  0

Every finite planar graph admits a proper vertex coloring with at most five colors. Euler formula for a connected planar graph supplies a vertex of degree at most five. Inductively color its deletion. The only obstruction is five differently colored neighbours in cyclic order. If opposite neighbours 1 and 3 are in different two-color components, interchange one component. If they are connected, planarity prevents opposite neighbours 2 and 4 from being connected in their two-color subgraph, and that interchange frees a color instead. This is a Kempe chain proof.

Every finite planar graph has chromatic number at most four. This stronger theorem is distinct from the elementary five colour theorem; the latter's Kempe chain proof does not prove this assertion.

For a connected plane drawing, Euler formula for a connected planar graph isFor a simple planar graph with $v\ge 3$, counting edge sides around faces gives $3f\le 2e$ and hence $e\le 3v-6$. Thus some vertex has degree at most five. A tree or a graph with bridges obeys the same bound; a face boundary is counted as a closed walk with edge multiplicity, and the small cases are immediate.

Induct on the number of vertices to prove the five colour theorem. Remove a vertex $v$ of degree at most five and five-colour the remainder. If fewer than five colors appear at its neighbours, extend the coloring. Otherwise it has five neighbours in their cyclic order, with colors $1,2,3,4,5$. If the color-1 neighbour is not connected to the color-3 neighbour by a path using only colors 1 and 3, swap these colors in the component containing the former, freeing color 1 at $v$. If they are connected, that path together with its two edges to $v$ is a simple closed curve. The colour-2 and colour-4 neighbours lie on opposite sides, so cannot be joined by a path using only colors 2 and 4, which would have to cross it. Swap colors in the component containing the colour-2 neighbour and give $v$ color 2. This establishes the induction without the four colour theorem.

For a triangle-free simple planar graph with $v\ge 3$, the analogous bound is $e\le 2v-4$, so some vertex has degree at most three. One can prove the bound for connected graphs with no bridges by $4f\le 2e$; bridges and isolated components are then joined or split off, with the small tree cases checked directly. Every subgraph is still triangle-free and planar. Successively remove a vertex of degree at most three, and color in reverse order with four colors. Thus triangle-free planar graphs have chromatic number at most four.

Finally, let $G$ be minimal 5-chromatic. Removing any vertex leaves a graph colorable with four colors. If its degree is at most three, that coloring extends immediately. If its degree is four, either a colour is unused at its neighbours, or all four appear once in cyclic order. The same two-path argument, now using opposite pairs 1,3 and 2,4, permits a Kempe chain interchange freeing one colour. Therefore a planar minimal 5-chromatic graph must satisfyWithout planarity only $\delta (G)\ge 4$ follows from the elementary extension argument. The complete graph $K_5$ is a counterexample to the stronger assertion: it is minimal 5-chromatic and every vertex has degree four. Deleting an edge allows its endpoints to share a color; deleting a vertex also reduces the required number of colors.

Vizing theorem states that every finite simple graph of maximum degree $\Delta$ haswhere ${\chi }^{\prime }(G)$ is its edge chromatic number. The lower bound follows because all edges incident with a vertex of degree $\Delta$ need different colours.

For the upper bound, induct on the number of edges. Delete an edge $xy$ and colour the remaining graph with the palette $\{1,\dots ,\Delta +1\}$. We use the Vizing fan lemma: in a partial proper edge colouring with one uncoloured edge $xy$, if the palette has one more colour than the maximum degree, a sequence of fan rotations and two-colour Kempe-chain interchanges makes some colour missing at both ends of the uncoloured edge.

Here is the fan argument. Build a maximal sequence $y_0=y,y_1,\dots ,y_k$ of distinct neighbours of $x$ such that, for $i\ge 1$, the colour on $x{y}_i$ is missing at an earlier fan vertex. Rotating an initial fan segment means moving each colour on $x{y}_i$ to the preceding edge where it is missing; this preserves properness and moves the uncoloured edge to the end of that segment. Choose a colour $\alpha$ missing at $x$ and a colour $\beta$ missing at $y_k$. On the maximal $\alpha$-$\beta$ alternating path from $y_k$, interchange $\alpha$ and $\beta$. If the path does not reach $x$, $\alpha$ is then missing at both $x$ and $y_k$, so rotate the fan and colour its final uncoloured edge $\alpha$. If it reaches $x$, its last edge at $x$ has colour $\beta$. Maximality of the fan forces the other endpoint of that edge to be a fan vertex; rotate up to that vertex first. This breaks the alternating path before $x$, reducing to the preceding case. This proves the fan lemma and extends the colouring to $xy$, completing the induction.

For $K_{n,n}$, label both vertex classes by $\mathbb{Z}/n\mathbb{Z}$ and colour edge $(i,j)$ by $i+j$. Each colour is a perfect matching, so this is an $n$-edge-colouring. Since $\Delta (K_{n,n})=n$, ${\chi }^{\prime }(K_{n,n})=n$. If $r=\max \{m,n\}$, embed $K_{m,n}$ into $K_{r,r}$ and restrict its $r$-edge-colouring. The degree lower bound then gives

Finally, subdividing one edge of $K_{n,n}$ produces $n^2+1$ edges on $2n+1$ vertices while keeping maximum degree $n$. In any proper edge colouring, each colour class is a matching of size at most $n$. An $n$-edge-colouring could therefore colour at most $n^2$ edges, a contradiction. Thus ${\chi }^{\prime }(G)>n$, while Vizing theorem gives ${\chi }^{\prime }(G)\le n+1$, so

In a partial proper edge colouring with one uncoloured edge incident with a vertex $x$, a Vizing fan is a sequence of neighbours of $x$ whose incident edge colours are missing at earlier fan vertices. With one more colour than the maximum degree, fan rotations and a two-colour Kempe chain free one common colour at the endpoints of the uncoloured edge.