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The average vertex degree of a graph is the sum of its vertex degrees divided by its order. Double counting edge endpoints gives $\overline{d}(G)=2e(G)/|V(G)|$. It lies between the minimum degree of a graph and the maximum degree of a graph.

If a bipartite graph $H$ has $k$ vertices and maximum degree $d$, thenA dependent random choice argument finds, in one colour, enough vertices whose every subset of at most $d$ vertices has a large common neighbourhood; a greedy embedding then places $H$.

The chromatic index $\chi \prime (G)$, also called the edge chromatic number, is the least number of labels in a proper edge colouring of a finite undirected graph without self-loops. Equivalently, it is the smallest number of matchings in a graph whose set partition covers all the edges. We take $\chi \prime (G)=0$ when there are no edges. For an arbitrary edge colouring, one colour always suffices whenever an edge exists, so properness is essential to this definition. The chromatic number $\chi (G)$ instead colours vertices.^[ref]

To establish an exact chromatic index, supply both a proper edge colouring and a reason that fewer colours cannot work. Every edge at a fixed vertex must have its own colour, giving the lower bound $\chi \prime (G)\ge \Delta (G)$, where $\Delta (G)$ is the maximum degree of a graph. The bound need not be attained: it counts a single resource's workload, while conflicts can also accumulate around a graph cycle.

For the triangle in a graph $K_3$ with edges $ab,bc,ca$, every pair of edges shares a vertex, so all three require different labels. The assignment $c(ab)=1,c(bc)=2,c(ca)=3$ works, proving $\chi \prime (K_3)=3$ although $\Delta (K_3)=2$. In its line graph, those three edges become a three-vertex clique. Treating this as three pair-resource jobs makes the obstruction tangible: each resource has only two jobs, but every pair of jobs conflicts, so no slot can contain two jobs.

For the square cycle graph $C_4$ with edges $ab,bc,cd,da$, use $c(ab)=c(cd)=1$ and $c(bc)=c(da)=2$. Each colour class is a matching in a graph, and each vertex has one edge of each colour. Since $\Delta (C_4)=2$, this proves $\chi \prime (C_4)=2$. More generally, alternation closes on any even cycle graph $C_{2m}$, $m\ge 2$. On an odd cycle $C_{2m+1}$, $m\ge 1$, two-colour alternation would make the last and first edges share a colour at their common vertex; a third colour on the last edge suffices. Thus even cycle graphs have chromatic index two and odd cycle graphs have chromatic index three. These exact examples show why maximum degree of a graph alone does not determine the answer.

For every finite simple graph, Vizing's theorem givesHere $\Delta (G)$ is the maximum degree of a graph, taken as zero when there are no vertices. A bipartite graph has the stronger equality $\chi \prime (G)=\Delta (G)$. The six-job complete bipartite graph above attains this lower bound, while the triangle in a graph attains the extra-colour alternative. The simple-graph restriction in the upper bound matters.^[ref]

The bipartite graph equality still holds for finite loopless multigraphs, with parallel edges counted separately in each vertex degree. For instance, $r$ parallel jobs between two resources need exactly $r$ slots: every pair conflicts, and $\Delta =\chi \prime =r$. For a general finite loopless multigraph, however, the simple graph upper bound can fail. Put two parallel edges on each side of a triangle in a graph. All six edges pairwise share a vertex, so each needs its own colour; every vertex has degree of a vertex four. Consequently $\chi \prime =6>\Delta +1=5$. A multiplicity-aware bound is $\chi \prime (G)\le \Delta (G)+\mu (G)$, where $\mu (G)$ is the largest number of parallel edges between a pair of vertices. The doubled triangle in a graph has $\mu =2$ and attains that bound. These statements exclude self-loops; the two-distinct-resource scheduling model excludes them too.^[ref]

For historical orientation, Graph Theory by Reinhard Diestel (Colouring preview, §5.3, p. 132) dates the bipartite equality to 1916 and Vizing's theorem to 1964. These are textbook attributions; the original papers have not been inspected here. The results state existence of optimal or nearly optimal proper edge colourings, rather than saying that arbitrary greedy choices will find them. The list-chromatic index asks the further question of success for every sufficiently large family of allowed-colour lists.

A bipartite graph has equal list-chromatic index, chromatic index and maximum degree of a graph $\Delta$. Start with a proper $\Delta$-edge coloring. Orient its line graph toward smaller colors at one part and toward larger colors at the other. An edge of color $c$ has at most $(c-1)+(\Delta -c)=\Delta -1$ outgoing neighbours. Every induced orientation has a digraph kernel obtained from a stable matching in a bipartite graph with those preferences. Apply the kernel list-coloring lemma.

For a finite simple graph, choose an order of its vertices and process them one at a time. Give each vertex the smallest positive colour label not already used by its coloured neighbours, and keep all earlier assignments. This is greedy coloring: each step makes a locally available choice. The resulting graph coloring is proper because, for each edge, its later endpoint avoids the colour of its earlier endpoint.^[ref]

If the maximum degree of a graph is $\Delta (G)$, a vertex has at most $\Delta (G)$ previously coloured neighbours, forbidding at most that many distinct colours. At least one label among $1,\dots ,\Delta (G)+1$ remains available. Consequently every order givesThis is an upper bound, not a promise that greedy coloring attains the chromatic number.^[ref]

Consider the path graph $P_4$ with edges $12,23,34$. In the order $1,2,3,4$, greedy coloring assigns the colours $1,2,1,2$ respectively. In the order $1,4,2,3$, it assigns colours $1,1,2,3$ in processing order: vertices $1,4$ both get colour $1$, then vertex $2$ gets colour $2$, and finally vertex $3$ sees colours $2$ and $1$ at its two neighbours and must take colour $3$. Written in vertex order $1,2,3,4$, this second colouring is $1,2,3,1$.

The path graph has chromatic number two: the first assignment achieves two colours, and any edge rules out one. Its maximum degree of a graph is two, so three colours in the second assignment still satisfy the bound. The extra colour comes entirely from the processing order. The crown graph gives a larger family where a poor order can make greedy coloring use arbitrarily many colours although the chromatic number remains two.

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

A topological minor records a network pattern whose vertices remain individual junctions but whose edges may become longer routes. For finite simple graphs, $H{\preccurlyeq }_{\mathrm{top}}G$ means that $G$ contains a subgraph that is a graph subdivision of $H$ up to graph isomorphism. Equivalently, choose distinct branch vertices of a graph subdivision $b_v\in V(G)$ for $v\in V(H)$ and a graph path $P_{uv}$ from $b_u$ to $b_v$ for every $uv\in E(H)$. These are internally vertex-disjoint paths, and no internal vertex of any $P_{uv}$ is a branch vertex of a graph subdivision. Only endpoints corresponding to a shared target vertex may be shared.^[ref]

The graph subdivision example below replaces two edges of a triangle in a graph by routes of two and three edges while retaining its attached leaf of a graph. If the host graph also contains an extra edge $BD$, the same routes still witness a topological minor: delete $BD$ when selecting the subgraph. An induced subgraph is not required. However, routes that merely connect the right pairs are insufficient: using a branch vertex of a graph subdivision as an intermediate junction, or making two routes share an internal vertex, can change the local network structure.

Every topological minor is a graph minor. In each branch path of a graph subdivision, contract all but one edge; the disjoint interiors let this shorten each route without merging distinct branch vertices of a graph subdivision. This recovers $H$ after deleting unused vertices and edges.^[ref] The converse fails because an ordinary branch set of a graph minor can gather several junctions into one.

For the tree $T$ with vertices $u,v,a,b,c,d$ and edges $uv,ua,ub,vc,vd$, contracting $uv$ produces a star graph $K_{1,4}$: the branch sets of a graph minor are $\{u,v\},\{a\},\{b\},\{c\},\{d\}$. Both $u,v$ have degree of a vertex three; each of the other four vertices is a leaf of a graph with degree of a vertex one. The new centre $w$ has four distinct graph neighbours $a,b,c,d$; its degree of a vertex is $3+3-2=4$, in agreement with the edge contraction formula.

Yet $K_{1,4}$ is not a topological minor of $T$. In a graph subdivision of $K_{1,4}$, four branch paths of a graph subdivision must leave the centre through four different edges; sharing their first internal vertex would violate internal disjointness. Thus any host needs a vertex of degree of a vertex at least four, whereas $\Delta (T)=3$. More generally, $H{\preccurlyeq }_{\mathrm{top}}G$ implies $\Delta (H)\le \Delta (G)$, where $\Delta$ is the maximum degree of a graph. This bound concerns the host graph; extra unused edges may make its branch vertices of a graph subdivision have larger degrees of vertices than their target counterparts.

This distinction matters when a network's individual junctions have a limited number of connections: replacing a connection by a route preserves the number of routes meeting at each chosen junction, while contracting a connected region can combine connections from separate junctions. It also explains why a small nonplanar pattern can hide inside long routes. By Kuratowski's theorem, a finite simple graph is a planar graph exactly when it has neither the complete graph $K_5$ nor the complete bipartite graph $K_{3,3}$ as a topological minor. The obstruction is a graph subdivision inside the network, regardless of the number of intermediate vertices.