Chromatic Index (CRX)

• Chromatic Index (CRX) is defined as the minimum number of colors needed to properly color a graph’s edges, ensuring that no adjacent edges share the same color.
• The theory distinguishes graphs into Class 1 and Class 2 based on whether the chromatic index equals the maximum degree or exceeds it by one, highlighting key structural properties.
• Extensions of CRX, including fractional, strong, conflict‐free, and modular variants, offer refined perspectives for addressing complex edge-colouring challenges in diverse graph settings.

The chromatic index, usually denoted $\chi'(G)$, is the smallest number of colours needed to colour the edges of a graph $G$ so that adjacent edges receive different colours. Formally, for a finite simple graph $G$, $$\chi'(G)=\min\{\,q:\text{there is a proper edge colouring of }G\text{ with }q\text{ colours}\,\}.$$ In the literature surveyed here, this ordinary edge-chromatic number is often the default meaning of “chromatic index,” but closely related edge-colouring parameters also appear, including the strong chromatic index, the fractional chromatic index, the $\bmod k$ chromatic index, the conflict-free chromatic index, and the distinguishing chromatic index. This suggests that “Chromatic Index (CRX)” functions both as a classical invariant and as a hub for several distance-constrained, arithmetic, fractional, and symmetry-sensitive refinements (Bruhn et al., 2016).

1. Foundational definition and equivalent formulations

For a simple graph $G=(V,E)$, a proper edge colouring assigns colours to edges so that no two adjacent edges share a colour, and $\chi'(G)$ is the minimum number of colours in such a colouring (Bruhn et al., 2016). In this form, the parameter is the edge analogue of the ordinary vertex chromatic number. The basic degree constraint is immediate: if $\Delta(G)$ denotes the maximum degree, then every proper edge colouring requires at least $\Delta(G)$ colours, because all edges incident with a maximum-degree vertex must be pairwise distinct.

A central classical theorem states that for any simple graph, $\chi'(G)\in\{\Delta(G),\Delta(G)+1\}$ (Bruhn et al., 2016). This yields the standard dichotomy into Class 1 graphs, with $G$0, and Class 2 graphs, with $G$1 (Bruhn et al., 2016). In multigraphs, the analogous general upper bound recorded in the surveyed literature is $G$2, where $G$3 is the maximum multiplicity (Chen et al., 2016). The same source also records the standard inequality chain

$G$4

placing the integral chromatic index between maximum degree and its fractional relaxation (Chen et al., 2016).

In some settings, the edge-colouring problem is recast through auxiliary graphs. For strong edge-colouring, the strong chromatic index satisfies $G$5, where $G$6 is the line graph and $G$7 its square (Huang et al., 2018). More broadly, the distance-$G$8 chromatic index is defined by requiring distinct colours on edges at distance at most $G$9, and satisfies $G$0 (Kaiser et al., 2012). This suggests that a large part of CRX theory can be viewed as vertex colouring of graph powers derived from line graphs.

A common misconception is that all parameters called “chromatic index” are variations of proper edge-colouring in the usual sense. The surveyed papers show otherwise. Some retain the usual adjacency constraint, such as the ordinary chromatic index $G$1 (Bruhn et al., 2016). Others change the local condition entirely, as in the $G$2 chromatic index, where each colour class must induce a subgraph all of whose non-isolated vertex degrees are congruent to $G$3 (Botler et al., 2022), or the conflict-free chromatic index, where the closed neighbourhood of every edge must contain a uniquely coloured edge (Kamyczura et al., 2022).

2. Structural theorems for ordinary chromatic index

A major direction in CRX theory is to identify structural graph classes for which $G$4. One such line concerns bounded treewidth. A conjecture of Vandenbussche, Sanders, Zhao, and Wehlau states that any graph of treewidth $G$5 and maximum degree $G$6 has chromatic index $G$7 (Bruhn et al., 2016). In support of this, the paper proves its fractional version and also proves that any graph of treewidth $G$8 and maximum degree $G$9 satisfies $$\chi'(G)=\min\{\,q:\text{there is a proper edge colouring of }G\text{ with }q\text{ colours}\,\}.$$0, improving an old result of Vizing (Bruhn et al., 2016). A central technical ingredient is the edge bound

$$\chi'(G)=\min\{\,q:\text{there is a proper edge colouring of }G\text{ with }q\text{ colours}\,\}.$$1

for graphs of treewidth $$\chi'(G)=\min\{\,q:\text{there is a proper edge colouring of }G\text{ with }q\text{ colours}\,\}.$$2 and maximum degree $$\chi'(G)=\min\{\,q:\text{there is a proper edge colouring of }G\text{ with }q\text{ colours}\,\}.$$3 (Bruhn et al., 2016).

Chordless graphs provide a different example of a structural class with especially strong control. For every chordless graph $$\chi'(G)=\min\{\,q:\text{there is a proper edge colouring of }G\text{ with }q\text{ colours}\,\}.$$4 with maximum degree $$\chi'(G)=\min\{\,q:\text{there is a proper edge colouring of }G\text{ with }q\text{ colours}\,\}.$$5, the strong chromatic index satisfies $$\chi'(G)=\min\{\,q:\text{there is a proper edge colouring of }G\text{ with }q\text{ colours}\,\}.$$6 (Basavaraju et al., 2013). The proof uses the fact that chordless graphs with $$\chi'(G)=\min\{\,q:\text{there is a proper edge colouring of }G\text{ with }q\text{ colours}\,\}.$$7 satisfy $$\chi'(G)=\min\{\,q:\text{there is a proper edge colouring of }G\text{ with }q\text{ colours}\,\}.$$8, together with a structural analysis of contracted matching graphs $$\chi'(G)=\min\{\,q:\text{there is a proper edge colouring of }G\text{ with }q\text{ colours}\,\}.$$9, which are shown to be 2-degenerate and hence 3-colourable (Basavaraju et al., 2013). This suggests that, in some sparse hereditary classes, ordinary CRX behaviour and stronger distance-2 edge-colouring behaviour can be controlled simultaneously by local structural decompositions.

Strongly regular graphs furnish a dense and highly symmetric setting. For connected strongly regular graphs of even order, all investigated examples except the Petersen graph are class 1, and the authors conjecture that this holds for all connected strongly regular graphs of even order (Cioaba et al., 2018). They determine the chromatic index of many strongly regular graphs, including all primitive strongly regular graphs of degree $\bmod k$0 and their complements, the Latin square graphs and their complements, and the triangular graphs and their complements (Cioaba et al., 2018). They also show asymptotically that a primitive strongly regular graph of even order, which is not the block graph of a Steiner 2-design or its complement, is class 1 for large enough order (Cioaba et al., 2018).

Dense quasirandom graphs supply another setting in which ordinary CRX becomes highly rigid. For lower-$\bmod k$1-regular graphs with $\bmod k$2, Glock, Kühn and Osthus proved the overfull criterion for even order, and Shan proved the odd-order analogue (Shan, 2021). In particular, for dense quasirandom graphs of odd order satisfying the same degree-spread condition,

$\bmod k$3

(Shan, 2021). This is an affirmative quasirandom version of the Chetwynd–Hilton conjecture in both parity cases.

3. Fractional chromatic index and the Goldberg–Jakobsen programme

The fractional chromatic index $\bmod k$4 is defined by weighting matchings so that every edge receives total weight 1, and minimizing the total weight (Bruhn et al., 2016). It satisfies $\bmod k$5, and in many contexts it is easier to analyze than $\bmod k$6 because it is the optimum of a linear program associated with the matching polytope (Bruhn et al., 2016). When $\bmod k$7, Edmonds’ theorem yields the exact formula

$\bmod k$8

and one may restrict to induced odd-order subgraphs (Chen et al., 2016).

This leads directly to Goldberg’s conjecture. A graph is called elementary if $\bmod k$9, and the conjecture asserts that if $G=(V,E)$0, then $G=(V,E)$1 is elementary (Chen et al., 2016). The paper “Chromatic index determined by fractional chromatic index” proves that if

$G=(V,E)$2

then $G=(V,E)$3 is elementary, improving the previous best square-root gap to a cube-root gap (Chen et al., 2016). It also verifies Jakobsen’s conjecture for all $G=(V,E)$4, and consequently shows that Goldberg’s conjecture holds for graphs with $G=(V,E)$5 or $G=(V,E)$6 (Chen et al., 2016).

In bounded-treewidth graphs, the fractional theory reaches the conjectured threshold. If $G=(V,E)$7 has treewidth $G=(V,E)$8 and maximum degree $G=(V,E)$9, then $\chi'(G)$0 (Bruhn et al., 2016). The proof proceeds by showing such graphs are not overfull, using the edge bound above and the monotonicity of treewidth under taking subgraphs (Bruhn et al., 2016). A plausible implication is that, in several dense or decomposable graph classes, the main obstruction to $\chi'(G)$1 is already visible at the fractional level through overfull subgraphs and odd-density inequalities.

4. Strong and distance-constrained chromatic indices

The strong chromatic index $\chi'(G)$2 is the minimum number of colours in an edge-colouring such that if two edges receive the same colour, they are neither incident nor incident with a common edge (Huang et al., 2018). Equivalently, every colour class induces a matching in $\chi'(G)$3, and more precisely an induced matching (Huang et al., 2018). The central conjecture here is the Erdős–Nešetřil conjecture: $\chi'(G)$4 This bound is best possible if true, via blow-ups of $\chi'(G)$5 (Huang et al., 2018).

For maximum degree four, the conjectured sharp bound is 20, and the paper “Strong chromatic index of graphs with maximum degree four” proves

$\chi'(G)$6

for every graph with maximum degree four, improving Cranston’s earlier bound 22 (Huang et al., 2018). The proof uses a minimal-counterexample argument, establishes 4-regularity, absence of small edge-cuts, and girth at least 6, and then constructs a global partition $\chi'(G)$7 that allows the colourings of $\chi'(G)$8 and $\chi'(G)$9 to be combined without conflict (Huang et al., 2018). This is markedly different from the probabilistic methods used for large $\Delta(G)$0.

For chordless graphs, a linear bound is possible: every chordless graph $\Delta(G)$1 with maximum degree $\Delta(G)$2 satisfies $\Delta(G)$3, and this is tight up to an additive constant (Basavaraju et al., 2013). For subcubic planar multigraphs, the strong chromatic index is at most 9, and this is sharp (Kostochka et al., 2015). These results show that strong edge-colouring behaves very differently in sparse hereditary classes than in the unrestricted setting, where the best known general upper bounds remain quadratic in $\Delta(G)$4 (Huang et al., 2018).

The distance-$\Delta(G)$5 chromatic index $\Delta(G)$6 generalizes both ordinary and strong chromatic index by requiring distinct colours on edges at distance at most $\Delta(G)$7. It satisfies

$\Delta(G)$8

and the trivial bound $\Delta(G)$9 (Kaiser et al., 2012). Kaiser and Kang prove that for every fixed $\Delta(G)$0, there exists an absolute $\Delta(G)$1 such that

$\Delta(G)$2

for sufficiently large $\Delta(G)$3 (Kaiser et al., 2012). They also prove that if the girth of $\Delta(G)$4 is at least $\Delta(G)$5, then

$\Delta(G)$6

and this is tight up to a constant multiplicative factor (Kaiser et al., 2012). This suggests that high girth converts the line-graph power into a graph with sparse neighborhoods, which is then amenable to Johansson–Molloy–Reed–AKS type colouring arguments.

5. Specialized settings and arithmetic or geometric variants

Several papers treat chromatic index in highly specialized graph families where exact formulas are available. One example is the family of integral sum graphs $\Delta(G)$7. For these graphs, the authors compute exact formulas such as

$\Delta(G)$8

in the parameter ranges treated, and compare them to an edge-sum chromatic index that is always one larger in those families (R et al., 2023). They call such graphs non-perfect edge-sum color graphs (R et al., 2023).

In finite geometry, the chromatic index of projective space $\Delta(G)$9 is defined by colouring lines so that intersecting lines receive different colours (Xu et al., 2022). The paper shows that for odd $\chi'(G)\in\{\Delta(G),\Delta(G)+1\}$0 and $\chi'(G)\in\{\Delta(G),\Delta(G)+1\}$1,

$\chi'(G)\in\{\Delta(G),\Delta(G)+1\}$2

which implies the existence of a parallelism of $\chi'(G)\in\{\Delta(G),\Delta(G)+1\}$3 (Xu et al., 2022). More generally, it reduces the determination of $\chi'(G)\in\{\Delta(G),\Delta(G)+1\}$4 to questions about $\chi'(G)\in\{\Delta(G),\Delta(G)+1\}$5, $\chi'(G)\in\{\Delta(G),\Delta(G)+1\}$6, and a special spread property called Property E (Xu et al., 2022).

In design theory, the chromatic index of a Steiner triple system $\chi'(G)\in\{\Delta(G),\Delta(G)+1\}$7 is the smallest number of partial parallel classes into which the triples can be partitioned (Bryant et al., 2017). For $\chi'(G)\in\{\Delta(G),\Delta(G)+1\}$8, the paper constructs Steiner triple systems with chromatic index at least $\chi'(G)\in\{\Delta(G),\Delta(G)+1\}$9 for each $G$00, with four possible exceptions, and proves that the maximum number of disjoint parallel classes in the systems constructed is sublinear in $G$01 (Bryant et al., 2017). It also shows that for each order $G$02 there are at least

$G$03

non-isomorphic Steiner triple systems with chromatic index at least $G$04, and that some of these systems are cyclic (Bryant et al., 2017).

Signed multigraphs require a different formalism: colours come from symmetric sets with self-inverse and paired elements, and the chromatic index $G$05 extends ordinary edge-colouring (Steffen et al., 2022). In this setting, Shannon’s theorem extends as

$G$06

and if $G$07 is balanced, then

$G$08

Moreover, $G$09 if and only if $G$10 has a matching $G$11 such that $G$12 is even (Steffen et al., 2022). This shows that balance plays a role analogous to a structural parity condition in the signed setting.

6. Alternative CRX notions: modular, conflict-free, and symmetry-sensitive

The $G$13 chromatic index $G$14 replaces the properness condition by a congruence condition: each colour class must span a subgraph in which every non-isolated vertex has degree congruent to $G$15 (Botler et al., 2022). Every proper edge-colouring is a $G$16-colouring, so $G$17 (Botler et al., 2022). For random graphs, the parameter becomes essentially constant: if $G$18 and $G$19, then asymptotically almost surely $G$20 when $G$21 is odd, and for even $G$22, $G$23 while $G$24 (Botler et al., 2022). The same paper records general bounds $G$25, improved via later work to $G$26, and a lower-bound example $G$27 (Botler et al., 2022).

The conflict-free chromatic index $G$28 is defined so that the closed neighbourhood of every edge contains a uniquely coloured edge (Kamyczura et al., 2022). Earlier work gave $G$29 with $G$30, while complete graphs yield lower bounds of order $G$31 (Kamyczura et al., 2022). The note “A note on the conflict-free chromatic index” proves the explicit upper bound

$G$32

and hence

$G$33

for graphs without isolated vertices (Kamyczura et al., 2022). It also shows that for bipartite graphs $G$34 (Kamyczura et al., 2022). A related paper shows that the maximum possible conflict-free chromatic index among graphs with maximum degree $G$35 is $G$36, much smaller than the $G$37 extremal behaviour of the conflict-free chromatic number (Dębski et al., 2020).

The distinguishing chromatic index $G$38 asks for a proper edge-colouring preserved only by the identity automorphism (Alikhani et al., 2017). For connected graphs of order at least 3,

$G$39

except for $G$40, $G$41, $G$42, and $G$43, where $G$44 (Alikhani et al., 2017). The same paper shows that $G$45, with equality only for those four graphs, computes exact values for paths, cycles, complete graphs, complete bipartite graphs, trees, friendship graphs, and book graphs, and proves that for connected class 2 graphs, $G$46 (Alikhani et al., 2017). It also establishes the line-graph identity

$G$47

for connected $G$48 outside two small exceptions, linking edge symmetry-breaking to vertex symmetry-breaking on line graphs (Alikhani et al., 2017).

Taken together, these variants show that “Chromatic Index (CRX)” is not a single isolated invariant but a family of edge-colouring parameters shaped by the same core question: how many colours are needed once a chosen local or global constraint is imposed on edge colour classes. A plausible implication is that modern CRX research is best understood as a study of how graph structure—degree, density, tree-likeness, girth, geometry, design incidence, arithmetic labeling, or automorphism group—changes the obstruction set from simple adjacency to more specialized local conditions.