Computing the Ramsey Number $R(K_5-P_3,K_5)$

We give a computer-assisted proof of the fact that $R(K_5-P_3, K_5)=25$. This solves one of the three remaining open cases in Hendry’s table, which listed the Ramsey numbers for pairs of graphs on 5 vertices. We find that there exist no $(K_5-P_3,K_5)$-good graphs containing a $K_4$ on 23 or 24 vertices, where a graph $F$ is $(G,H)$-good if $F$ does not contain $G$ and the complement of $F$ does not contain $H$. The unique $(K_5-P_3,K_5)$-good graph containing a $K_4$ on 22 vertices is presented.

💡 Research Summary

The paper determines the exact Ramsey number R(K5‑P3, K5) = 25 by means of a computer‑assisted proof, thereby closing one of the three remaining open entries in Hendry’s 1989 table of Ramsey numbers for all connected graphs on five vertices. The authors first enumerate the families of (K4‑P3, K5)-good and (K5‑P3, K4)-good graphs, obtaining 1 092 and 3 454 499 non‑isomorphic members respectively, using a vertex‑by‑vertex extension algorithm together with the nauty package for isomorphism rejection. These enumerations confirm earlier results and provide the necessary building blocks for the main argument.

A key observation is that any (K5‑P3, K5)-good graph on 25 vertices must contain a K4, because R(K4, K5) = 25. Selecting the vertex of smallest degree within such a K4, denoted x, the authors split the remaining vertices into the neighbourhood F⁺ₓ (the vertices adjacent to x) and the anti‑neighbourhood F⁻ₓ (the non‑adjacent vertices, excluding x). By Lemma 1 they show that the sum of the degrees of the four vertices of the K4 is at most n + 8, which yields tight bounds on |V(F⁺ₓ)| and |V(F⁻ₓ)| for each possible order n ≥ 22. These bounds are summarized in Table IIIⅠ.

The next step introduces the notion of “cones”: each vertex v in F⁺ₓ is associated with a set of vertices in F⁻ₓ to which v is adjacent. Four combinatorial constraints (C1–C4) are derived from the requirement that neither a K5‑P3 nor an independent set of size five may appear. Roughly, (C1) forces the cones of vertices forming a triangle in F⁺ₓ to be pairwise disjoint; (C2) forbids the complement of the union of cones of two non‑adjacent vertices from containing an independent triple; (C3) requires the complement of the union of cones of three mutually non‑adjacent vertices to be a complete graph; and (C4) prevents the intersection of cones of adjacent vertices from containing an edge. These constraints dramatically prune the search space for admissible cone configurations.

The authors then exhaustively search all feasible cone assignments for each candidate size of F⁻ₓ (from 14 up to 17 vertices). For n = 25, 24, and 23 the search yields no valid cone arrangement, proving that no (K5‑P3, K5)-good graph containing a K4 exists on those orders. When n = 22, exactly one arrangement satisfies all constraints: F⁺ₓ consists of a triangle together with two isolated vertices (i.e., C3 ∪ C4), and the corresponding cones produce a unique (K5‑P3, K5; 22)-good graph. Its adjacency matrix is displayed in Figure 2; vertices 1‑4 form the K4, vertices 5‑8 induce a C4, and vertices 9‑22 comprise the anti‑neighbourhood F⁻ₓ.

Beyond the main result, the paper connects the finding to a theorem of Burr, Erdős, Faudree, and Schelp concerning extensions of complete graphs. By adapting their theorem, the authors show that several related Ramsey numbers involving “bKₙ,ₚ” graphs (a Kₙ with a new vertex attached to p existing vertices) also equal 25, specifically R(K4, bK₅,₂), R(bK₄,₁, K₅), R(bK₄,₁, bK₅,₂), R(bK₄,₂, K₅), and R(bK₄,₂, bK₅,₂). This demonstrates that the difficulty of the present case (K5‑P3 versus K5) is substantially greater than that of the other extensions covered by the earlier theorem.

In conclusion, the authors provide a rigorous, computer‑verified proof that R(K5‑P3, K5) = 25, identify the exact structure of the unique 22‑vertex (K5‑P3, K5)-good graph containing a K4, and illustrate how their methodology can be applied to other small Ramsey‑number problems. The work not only resolves a long‑standing open case but also offers insight into the interplay between avoiding a larger forbidden subgraph and avoiding its smaller core, suggesting avenues for future research on Ramsey numbers of small graph extensions.