Rainbow Subgraphs in Star-Coloured Graphs

• The paper establishes new extremal bounds for star-colouring complete graphs to avoid rainbow subgraphs, offering both asymptotic and exact formulas.
• It employs constructions such as lexical, orientable, and Turán-type methods to achieve sharp rainbow colouring thresholds within structured color classes.
• The research connects rainbow subgraph decompositions to broader Ramsey theory and matroid frameworks, providing insights for further combinatorial optimization.

A rainbow subgraph in a star-coloured graph is a subgraph in which all edges receive distinct colours, and every colour class in the graph induces a star (i.e., is a star-colouring). This concept is central for various extremal and decomposition questions at the intersection of graph colouring theory and generalized Ramsey theory, particularly for understanding the structure and limits of edge-colourings that avoid rainbow appearances of prescribed target graphs or possess sharp combinatorial decomposability into rainbow spanning stars.

1. Fundamental Definitions and Structural Properties

An edge-colouring of a graph $G$ is a map $c: E(G) \to C$, where $C$ is a set of colours. A colouring is:

• Proper if it forbids monochromatic cherries (pairs of incident edges in the same colour).
• Star-colouring if it forbids monochromatic matchings of size two and triangles, so that each colour class forms a star (every edge of a given colour is incident to a unique vertex, and no two disjoint edges receive the same colour).

A rainbow subgraph is a subgraph in which every edge receives a unique colour. The primary object of paper is the extremal function $f(n,H)$, defined as the maximum number of colours in a star-colouring of the complete graph $K_n$ with no rainbow copy of a fixed graph $H$ (Lo et al., 16 Nov 2025).

Star-coloured graphs, especially those with spanning stars as colour classes, play a crucial role in the combinatorial decomposition problems and in the refinement of classical extremal results for rainbow subgraphs.

2. Extremal Functions and Main Theorems

Let $f(n, H)$ denote the maximal number of colours valid for a star-colouring of $K_n$ avoiding a rainbow $H$. This function connects to generalized Ramsey numbers in the mixed Ramsey framework. The asymptotic and exact values of $f(n, H)$ depend critically on the vertex arboricity $a(H)$ of $H$ (Lo et al., 16 Nov 2025):

• If $a(H) \geq 3$, then

$$f(n, H) = (1+o(1)) \bigg(1 - \frac{1}{a(H)-1}\bigg) \binom{n}{2}.$$

Here, almost all edges can be uniquely coloured while avoiding rainbow $H$.

• For $a(H) = 2$, various exact formulas are obtained, including:
  • For cycles $C_k$ ($k \geq 3$):

  $f(n, C_k) = n + \binom{k-2}{2} - 1.$ - For the 4-clique $K_4$:

  $f(n, K_4) = 2n - 3.$ - For $K_4^-$ (the 4-clique with one edge removed):

  %%%%3%%%% - For the graph $K_5^-$ (join of $K_2$ and $P_2$):

  $$(1-o(1))(n/2)^{3/2} \leq f(n, K_5^-) \leq (1+o(1))16(n/2)^{3/2}.$$ - For joins of trees $H = T_1 + T_2$ with $|T_1| = s$:

  $f(n, H) = \Theta(n^{2-1/s}).$

These extremal results refine and extend previous general Ramsey-type results, with sharper control in the star-colouring regime.

3. Constructions and Extremal Colouring Methods

Several canonical constructions realise these bounds and exhibit the extremal nature of star-colourings:

• Lexical/Transitive Star-Colouring: Vertices are linearly ordered, and each edge $v_i v_j$ is coloured by $i$. This uses $n-1$ colours and avoids any rainbow cycle.
• Orientable Colourings: By constructing a tournament on the vertex set and colouring arcs in each out-star with the colour of the centre, star-colourings with $n$ or $n-1$ colours are realised.
• Rainbow Blow-Up (Turán-type): Partition the vertex set into $\ell$ parts, star-colour each part, and assign new colours to crossing edges, achieving $$f(n,H) \approx t_{\ell}(n) + (\text{colours inside parts})$$.
• Sparse Rainbow Boost: By recolouring a sparse subgraph rainbow and ensuring every copy of $H$ meets this subgraph, the overall colour count can be increased without introducing unwanted rainbow subgraphs.

These constructions clarify both lower and upper bounds for $f(n,H)$ and demonstrate the combinatorial tightness of the obtained extremal values (Lo et al., 16 Nov 2025).

4. Upper Bound Techniques and Structural Lemmas

Upper bounds on $f(n,H)$ employ combinatorial induction and structural decompositions:

• In the $K_4$ case, induction on $n$ is used along with an analysis of vertices of high star-centre degree. Sophisticated bipartitions of the vertex set and color absorption mechanisms ensure that all colours are confined and no rainbow $K_4$ can arise.
• For cycles and $K_4^-$, the structure of unique-colour edges is exploited, showing that they must form a matching plus at most one 2-path.
• For larger graphs such as $K_5^-$ or joins of trees, techniques from extremal graph theory such as Zarankiewicz-type arguments and dependent random choice are applied to obtain optimal bounds.

Proofs frequently combine classic extremal graph concepts with star-colouring-specific combinatorial constructions, distinguishing the star-colouring regime from the classical proper-colouring setting (Lo et al., 16 Nov 2025).

5. Decompositions into Rainbow Stars and Connections to Matroid Theory

The structure of star-coloured graphs underpins decomposition problems relevant to matroid theory. When the colour classes of a graph induce monochromatic spanning stars, sharp necessary and sufficient conditions exist for the decomposition of the graph into edge-disjoint rainbow stars:

• If all stars have distinct centres or all share the same centre, a decomposition into rainbow stars is always possible.
• For two centres, a decomposition into rainbow spanning trees is possible, though not always into stars.
• In all other cases, such a decomposition is not possible—degree counting confirms that $s_x \in \{0,1,n-1\}$ for each vertex $x$ (number of times $x$ appears as a centre) is necessary and sufficient (Asthana et al., 2023).

These results are deeply connected to Rota's Basis Conjecture in graphic matroids and rely on tools such as greedy matching, Latin square analysis, and double rotation schemes.

6. Broader Ramsey Framework and Implications

The paper of rainbow subgraphs in star-coloured graphs is a distinguished case in the general theory of mixed Ramsey numbers. The function $f(n,H)$ is a refinement of $R_{\{M_2, K_3\}}(n, H)$, with sharper asymptotics determined by the vertex arboricity:

• For $a(H) \geq 3$, the star-colouring setting matches the known (up to lower order terms) density bounds.
• For lower arboricity or “smaller” target graphs, the function exhibits diverse behaviour with polynomial exponents explicitly realised by various graph classes.
• The results solve open extremal questions for cycles, small cliques, incomplete cliques, and joins of trees, accompanied by explicit extremal constructions and decompositions (Lo et al., 16 Nov 2025).

These considerations position the star-colouring and rainbow subgraph framework as a robust extension of classical Ramsey theory and a promising setting for further extremal investigations.