Reconfiguration graph for vertex colorings for ($P_2$+$P_3$, $C_4$)-free graphs

For a graph $G$, let $χ(G)$ denote the chromatic number of $G$. Given a graph $G$, the $reconfiguration$ $graph$ $for$ $the$ $k$-$colorings$ of $G$, denoted by ${\cal R}_k(G)$, is the graph whose vertices are the $k$-colorings of $G$ and two $k$-colorings are joined by an edge if they differ on exactly one vertex of $G$. A graph $G$ is $k$-$mixing$ if ${\cal R}_k(G)$ is connected, and is $recolorable$ if it is $k$-mixing for all $k> χ(G)$. In this paper, we give a complete characterization of $(P_2+P_3, C_4)$-free graphs that are recolorable. Moreover, we show that if $G$ is a recolorable $(P_2+P_3, C_4)$-free graph, then for any $k >χ(G)$, the diameter of ${\cal R}_k(G)$ is at most 2$n^{2}$. Furthermore, we show that if $G$ is a ($P_2+P_3, C_4$)-free graph on $n$ vertices with degeneracy $ρ(G)$, then for all $k > ρ(G)+ 1$, the diameter of ${\cal R}_k(G)$ is at most $O(n^2)$. This confirms a conjecture of Cereceda for the class of ($P_2+P_3, C_4$)-free graphs. These results generalize some known results available in the literature.

💡 Research Summary

The paper investigates the reconfiguration graph Rₖ(G) of k‑colorings for graphs G, focusing on the hereditary class of (P₂ + P₃, C₄)‑free graphs. After recalling basic definitions—k‑mixing, frozen colorings, and the Cereceda conjecture concerning the diameter of Rₖ(G) when k exceeds the graph’s degeneracy ρ(G)—the authors present a detailed structural analysis of the target class.

The core of the work is a two‑case structure theorem. In the first case, G contains a C₆; in the second, it does not. When G is C₆‑free, the authors use a partition of the vertex set relative to a C₅ (sets Aᵢ, Bᵢ, Dᵢ, Z, T) and prove a series of properties (O₁–O₈) that tightly control adjacency among these sets. If G also avoids a 5‑cap, the induced subgraph on C₅ ∪ D is a blow‑up of C₅; otherwise, the presence of a 5‑cap forces the existence of a pair of comparable vertices, which immediately yields k‑mixing for all k > χ(G).

When G contains a C₆, the authors introduce two new graph families, H₁ and H₂, defined by explicit configurations of a small number of distinguished vertices together with cliques that are either complete or anticomplete to them. They also allow the join of a blow‑up of C₅ with a complete graph Kₚ. The main theorem (Theorem 1) states that any connected (P₂ + P₃, C₄, C₆)‑free graph falls into one of three categories: (i) chordal, (ii) contains a pair of comparable vertices, or (iii) is the join of Kₚ with a graph from {blow‑up of C₅, H₁, H₂}. This structural decomposition is the key to the recoloring results.

Using this decomposition, the authors prove that every (P₂ + P₃, C₄)‑free graph that is recolorable (i.e., k‑mixing for all k > χ(G)) has a reconfiguration diameter bounded by 2 n² for any such k. The proof constructs a sequence of recolorings where each step changes the color of a single vertex, and the total number of steps never exceeds 2 n². This improves upon the generic O(n²) bound known for many hereditary classes by providing an explicit constant.

Moreover, they verify Cereceda’s conjecture for this class: if G is (P₂ + P₃, C₄)‑free with degeneracy ρ(G), then for all k > ρ(G)+1 the diameter of Rₖ(G) is O(n²). The proof leverages the fact that in the structural cases (i)–(iii) the graph either has low degeneracy or can be reduced to a blow‑up of a small cycle, both of which admit short recoloring sequences.

The paper also supplies several auxiliary lemmas (e.g., Lemma 1) that guarantee the existence of comparable vertices or describe the exact form of the graph when certain induced substructures are absent. These lemmas, together with the O₁–O₈ properties, form a toolbox that may be useful beyond the specific class studied.

In conclusion, the authors deliver a complete characterization of recolorable (P₂ + P₃, C₄)‑free graphs, establish tight quadratic diameter bounds for their reconfiguration graphs, and confirm the Cereceda conjecture for this family. The work extends known results for split and pseudo‑split graphs, introduces new graph families (H₁, H₂), and provides techniques that could be adapted to broader hereditary classes. Future research may explore whether similar structural decompositions and diameter bounds hold for larger forbidden subgraph families or for graphs that contain but avoid more complex configurations.