1/k-Majority (k+1)-Edge-Colouring in Graphs

• 1/k-majority (k+1)-edge-colouring is an edge-colouring that limits each colour to at most ⌊d(v)/k⌋ edges per vertex.
• The theory establishes sharp degree thresholds, with general graphs requiring δ(G) > 2k² and bipartite graphs needing δ(G) ≥ k(k–1).
• Innovative methods such as iterative decompositions, Eulerian splits, and rounding lemmas enable precise results and extend the framework to lists and hypergraphs.

A $\frac{1}{k}$-majority $(k+1)$-edge-colouring of a graph $G$ is an edge-colouring with $k+1$ colours such that, for every vertex $v$ and every colour $i$, at most $\left\lfloor \frac{d(v)}{k}\right\rfloor$ edges incident with $v$ receive colour $i$ (Pękała et al., 2023). Equivalently, no single colour may occupy more than a $\frac{1}{k}$-fraction of the edges incident with any vertex. This local density constraint generalizes the basic majority case $(k+1)$0, and the modern theory concerns the degree thresholds, extremal obstructions, proof methods, and extensions that determine when $(k+1)$1 colours suffice (Bock et al., 2022).

1. Definition and formal setting

The foundational notion is the majority $(k+1)$2-edge-colouring introduced for graphs as an edge-colouring $(k+1)$3 such that, for every vertex $(k+1)$4 and every colour $(k+1)$5,

$(k+1)$6

This is the case $(k+1)$7 in the more general $(k+1)$8-majority formulation, where one requires

$(k+1)$9

for every vertex $G$0 and colour $G$1 (Bock et al., 2022).

The specialized object of current interest is the case $G$2 together with exactly $G$3 colours. In the notation of the generalized theory, a $G$4-majority $G$5-edge-colouring is an edge-colouring with $G$6 colours such that for every colour $G$7 and each vertex $G$8, at most a $G$9-fraction of the edges incident with $k+1$0 have colour $k+1$1; for integer degrees this is written as

$k+1$2

The case $k+1$3 is the first nontrivial colour budget beyond $k+1$4, and it is the form around which the threshold conjecture and most sharp results are organized (Pękała et al., 2023).

A basic structural feature of the problem is that the condition is entirely vertexwise. It controls each colour class locally at every vertex rather than globally across the graph. This makes the problem closer in spirit to discrepancy theory and balanced decompositions than to classical proper edge-colouring.

2. Threshold conjecture and extremal obstructions

The central conjecture states that for every integer $k+1$5, if

$k+1$6

then $k+1$7 is $k+1$8-majority $k+1$9-edge-colourable (Pękała et al., 2023). The same source observes that such a result would be best possible.

There are two distinct reasons for this sharpness. First, $v$0 colours are necessary in general: with only $v$1 colours, a vertex whose degree is not divisible by $v$2 forces one colour to appear too many times, so no minimum-degree hypothesis can make $v$3 colours sufficient for all graphs (Pękała et al., 2023). Second, the conjectured degree threshold cannot be lowered below $v$4 in the general setting. For every $v$5, there exists a graph with minimum degree

$v$6

that is not $v$7-majority $v$8-edge-colourable; the construction starts with $v$9, deletes the edges of a Hamilton cycle, and adds a new vertex adjacent to all remaining vertices (Pękała et al., 2023).

The bipartite case has its own sharp obstruction. For every $i$0, there are bipartite graphs with minimum degree

$i$1

that are not $i$2-majority $i$3-edge-colourable; a concrete example is $i$4 with one edge removed (Pękała et al., 2023). This yields the exact bipartite threshold $i$5.

For the original majority case $i$6, the obstruction theory is more concrete. Any graph containing a vertex of degree $i$7 cannot have a majority edge-colouring at all. Moreover, there are class 2 graphs with minimum degree at least $i$8 and maximum degree $i$9 that do not have a majority $\left\lfloor \frac{d(v)}{k}\right\rfloor$0-edge-colouring. A further obstruction is identified: if $\left\lfloor \frac{d(v)}{k}\right\rfloor$1 has minimum degree at least $\left\lfloor \frac{d(v)}{k}\right\rfloor$2 and contains an induced subgraph $\left\lfloor \frac{d(v)}{k}\right\rfloor$3 such that $\left\lfloor \frac{d(v)}{k}\right\rfloor$4 is class 2 with maximum degree $\left\lfloor \frac{d(v)}{k}\right\rfloor$5, and all vertices of $\left\lfloor \frac{d(v)}{k}\right\rfloor$6 have degree $\left\lfloor \frac{d(v)}{k}\right\rfloor$7 or $\left\lfloor \frac{d(v)}{k}\right\rfloor$8 in $\left\lfloor \frac{d(v)}{k}\right\rfloor$9, then $v$0 has no majority $v$1-edge-colouring (Bock et al., 2022).

These lower bounds place the subject in a narrow asymptotic window. The conjectural threshold is quadratic in $v$2, and both the general and bipartite counterexamples show that this quadratic order is unavoidable.

3. Existence theorems for graphs

The principal graph-theoretic existence results form a progression from the basic majority case to the general $v$3-majority $v$4 setting (Bock et al., 2022, Pękała et al., 2023).

Setting	Minimum-degree condition	Conclusion
General graphs, $v$5	$v$6	majority $v$7-edge-colouring
General graphs, $v$8	$v$9	majority $i$0-edge-colouring
General graphs, arbitrary $i$1	$i$2	$i$3-majority $i$4-edge-colouring
General graphs, refined bound	$i$5	$i$6-majority $i$7-edge-colouring
Bipartite graphs	$i$8	$i$9-majority $\frac{1}{k}$0-edge-colouring
Small values	$\frac{1}{k}$1 for $\frac{1}{k}$2, $\frac{1}{k}$3 for $\frac{1}{k}$4	exact conjectured form for $\frac{1}{k}$5

In the basic majority case, the result that every graph with minimum degree at least $\frac{1}{k}$6 admits a majority $\frac{1}{k}$7-edge-colouring is stated to be best possible under the minimum-degree condition. Likewise, every graph with minimum degree at least $\frac{1}{k}$8 has a majority $\frac{1}{k}$9-edge-colouring, and this is again best possible under that hypothesis because a graph containing a vertex of odd degree at least $(k+1)$00 cannot have a majority $(k+1)$01-edge-colouring (Bock et al., 2022).

The 2023 advances substantially sharpen the general theory. They improve the previously known bound of order $(k+1)$02 to the direct non-random bound

$(k+1)$03

and then further to an asymptotic threshold of $(k+1)$04 (Pękała et al., 2023). The same work proves the conjecture itself for $(k+1)$05: the case $(k+1)$06 was already known, while $(k+1)$07 and $(k+1)$08 are established there.

The bipartite theorem is exact: if $(k+1)$09 is bipartite and

$(k+1)$10

then $(k+1)$11 has a $(k+1)$12-majority $(k+1)$13-edge-colouring, and the lower-bound example at degree $(k+1)$14 shows that this cannot be improved (Pękała et al., 2023). This sharp bipartite result is one of the cleanest parts of the theory.

4. Methods: Eulerian splits, rounding, and iterative decompositions

The earliest proofs already exhibit two important methodological themes. The strong $(k+1)$15-edge-colouring statement in the majority setting is proved using Euler tours, while the theorem that every graph of minimum degree at least $(k+1)$16 has a majority $(k+1)$17-edge-colouring uses the Gallai–Edmonds decomposition (Bock et al., 2022). These arguments establish that parity, local balance, and decomposition into structured subgraphs are central.

A later stage of the theory replaces probabilistic existence by a direct non-random mechanism. The key tool is Lemma 6, a refined decomposition lemma inspired by Alon–Wei. Given a graph $(k+1)$18 and weights

$(k+1)$19

it produces a function

$(k+1)$20

such that, for every vertex $(k+1)$21,

$(k+1)$22

It also has a stability property: if for an edge $(k+1)$23 both endpoint sums fall below the corresponding $(k+1)$24-targets, then necessarily $(k+1)$25. Finally, vertices with excess $(k+1)$26 are confined to pairwise independent odd cycles with integral local sums (Pękała et al., 2023).

This lemma is used iteratively to carve the edge set into $(k+1)$27 colour classes while controlling the number of edges of each colour incident with every vertex. In bipartite graphs the odd-cycle complication disappears, which explains why the bipartite threshold can be solved exactly. In general graphs, the residual odd-cycle structure prevents exact control but still allows the quadratic bound $(k+1)$28 and the improved $(k+1)$29 bound (Pękała et al., 2023).

The same paper supplements Lemma 6 with a flexible two-colouring tool derived from Eulerian structure. In one formulation, if a connected graph has an even number of edges or has a vertex of odd degree, then it admits a $(k+1)$30-edge-colouring such that each vertex sees at most $(k+1)$31 edges of one colour; in the even-edge, all-even-degree case, one vertex may be exceptional and receive one extra edge of a colour. This mechanism is used to finish residual subgraphs after part of the decomposition has already been performed (Pękała et al., 2023).

A notable conceptual point is the explicit limitation of straightforward probabilistic methods near the conjectured threshold. When $(k+1)$32, each colour should appear about $(k+1)$33 times around a vertex, while the allowed maximum is only $(k+1)$34. This means the colouring must control every colour count with additive error at most $(k+1)$35, and the 2023 paper argues that standard concentration tools appear too weak for that scale (Pękała et al., 2023). This observation motivates the shift from Lovász-local-lemma arguments to discrepancy-style iterative rounding.

5. List, infinite, and hypergraph extensions

The list version replaces a common colour set by an assignment $(k+1)$36 and asks for an $(k+1)$37-majority colouring chosen from the prescribed lists. For every integer $(k+1)$38, every graph with minimum degree

$(k+1)$39

admits a $(k+1)$40-majority edge colouring from any list assignment with lists of size $(k+1)$41. The same statement is extended to finite or infinite graphs (Pękała et al., 18 Feb 2025).

The proof is based on vertex splitting, Euler tours, and a reduction to bipartite list edge-colouring. Each high-degree vertex is split into copies of degree at most $(k+1)$42, the resulting components are made Eulerian, the Euler tours are oriented, and from the orientation one constructs a bipartite graph $(k+1)$43 of maximum degree at most $(k+1)$44. Galvin’s theorem is then applied to colour $(k+1)$45 from any lists of size $(k+1)$46, and the colouring is transferred back to the original graph. The counting argument shows that each vertex $(k+1)$47 is incident with at most

$(k+1)$48

edges of any fixed colour, which is at most $(k+1)$49 once $(k+1)$50 (Pękała et al., 18 Feb 2025). For $(k+1)$51, this settles the list version of the basic majority-edge-colouring conjecture.

The same work also studies more general list settings. If $(k+1)$52 and $(k+1)$53, then every graph with minimum degree

$(k+1)$54

has an $(k+1)$55-majority edge colouring from lists of size $(k+1)$56. It then extends to diversified tolerances $(k+1)$57 under the excessive condition

$(k+1)$58

with probabilistic existence theorems under explicit minimum-degree hypotheses (Pękała et al., 18 Feb 2025).

A further extension passes from graphs to hypergraphs. If $(k+1)$59 is a hypergraph of rank

$(k+1)$60

then for every integer $(k+1)$61 and $(k+1)$62, every hypergraph with minimum degree

$(k+1)$63

admits a $(k+1)$64-majority $(k+1)$65-edge-colouring (Ai et al., 26 Sep 2025). The proof first gives a weaker probabilistic threshold of order $(k+1)$66 via a weighted Lovász Local Lemma, and then improves it to $(k+1)$67 by extending the Pękała–Przybyło rounding lemma to hypergraphs. In the hypergraph version, the discrepancy error at a vertex becomes $(k+1)$68 rather than $(k+1)$69, reflecting the fact that one hyperedge can affect up to $(k+1)$70 vertex sums. The paper also proves that every linear hypergraph of maximum rank $(k+1)$71 and minimum degree at least $(k+1)$72 has a $(k+1)$73-majority $(k+1)$74-edge-colouring (Ai et al., 26 Sep 2025).

6. Related notions and distinctions

A recurring source of confusion is that not every “majority edge-colouring” in the literature uses the same local neighbourhood. The standard $(k+1)$75-majority $(k+1)$76-edge-colouring problem is vertex-based: the local counts are taken among edges incident with a vertex. By contrast, strong majority edge-colouring is edge-based. In that setting, for every edge $(k+1)$77 and every colour $(k+1)$78, at most half of the edges adjacent to $(k+1)$79 may have colour $(k+1)$80 (Antoniuk et al., 30 Jun 2026).

Formally, a strong majority edge-colouring of a graph $(k+1)$81 is an edge-colouring $(k+1)$82 such that for every edge $(k+1)$83 and every colour $(k+1)$84,

$(k+1)$85

The 2026 paper immediately rephrases this as: for every edge $(k+1)$86 and every colour $(k+1)$87, at most half of the edges adjacent to $(k+1)$88 have colour $(k+1)$89. This is naturally a $(k+1)$90-majority condition, but it is not the same problem as the vertexwise $(k+1)$91-majority $(k+1)$92 theory (Antoniuk et al., 30 Jun 2026).

The strong-majority setting has its own obstruction and colour bounds. Such a colouring exists only for admissible graphs, meaning graphs with no pendant path of length two. Kalinowski, Kamyczura, Pilśniak, and Woźniak proved that every admissible graph admits such a colouring with at most eight colours and conjectured that four colours always suffice; the 2026 result improves the general upper bound from $(k+1)$93 to $(k+1)$94. It also proves that every graph with no vertices of degree $(k+1)$95 or $(k+1)$96 satisfies the corresponding $(k+1)$97-colour bound, implying the same for every graph with $(k+1)$98. The proof uses the notion of being balanced at a vertex, equitable edge-colouring of Hilton and de Werra, and local recolouring on degree-$(k+1)$99 paths (Antoniuk et al., 30 Jun 2026).

Beyond graph edge-colouring, the broader majority paradigm includes vertex colourings of digraphs and majority partitions of $G$00-edge-coloured graphs. In digraphs, the extremal number $G$01 for $G$02-majority vertex colourings satisfies

$G$03

and every digraph is $G$04-majority $G$05-choosable (Girão et al., 2017). In $G$06-edge-coloured graphs, a majority partition requires that for each edge colour and each vertex, at least as many edges of that colour leave the vertex’s part as stay inside it; every such graph has a $G$07-majority partition, while deciding the existence of a $G$08-majority partition is NP-complete (Bang-Jensen et al., 26 Aug 2025). These frameworks are closely related in spirit, but they address different adjacency models and should not be conflated with $G$09-majority $G$10-edge-colouring of graphs.

Taken together, these results suggest a coherent picture. The vertexwise graph problem has a sharp quadratic threshold conjecture, exact bipartite behaviour, exact small-$G$11 solutions through $G$12, and strong list and hypergraph extensions. At the same time, related majority frameworks reveal that small changes in the local neighbourhood—vertices instead of edges, edges adjacent to an edge instead of edges incident with a vertex, or partitions instead of colourings—can change both the combinatorial obstructions and the optimal number of colours.