The Complexity of (P3, H)-Arrowing and Beyond

TL;DR

This work addresses the complexity of the fixed-variance Ramsey arrowing problem $(P_3,H)$, establishing a full coNP-hardness classification for all $2$-connected graphs $H$ with $|V(H)|\ge 4$ (except $H=K_3$, which is solvable in polynomial time). A key contribution is the introduction of the edge-pair linkage invariant $\mathrm{epl}_G(e,f)$ and its derived measure $\mathrm{mepl}(G)$, which enables safe amalgamation of graph copies via gadgets called enforcers and signal extenders. The authors develop SAT-based reductions from $(2,2)$-3SAT to $(P_3,H)$-Nonarrowing and extend the hardness to broader arrowing families, including $(P_3,TK_n)$ and $(K_{1,n},H)$, thereby connecting arrowing complexity to $H$-free Matching Removal. The results advance the program of classifying all $(F,H)$-Arrowing problems and provide a framework for constructing more intuitive hardness proofs without ad-hoc SAT gadgets, with implications for related Ramsey-type decision problems.

Abstract

Often regarded as the study of how order emerges from randomness, Ramsey theory has played an important role in mathematics and computer science, giving rise to applications in numerous domains such as logic, parallel processing, and number theory. The core of graph Ramsey theory is arrowing: For fixed graphs $F$ and $H$, the $(F, H)$-Arrowing problem asks whether a given graph, $G$, has a red/blue coloring of the edges of $G$ such that there are no red copies of $F$ and no blue copies of $H$. For some cases, the problem has been shown to be coNP-complete, or solvable in polynomial time. However, a more systematic approach is needed to categorize the complexity of all cases. We focus on $(P_3, H)$-Arrowing as $F = P_3$ is the simplest meaningful case for which the complexity question remains open, and the hardness for this case likely extends to general $(F, H)$-Arrowing for nontrivial $F$. In this pursuit, we also gain insight into the complexity of a class of matching removal problems, since $(P_3, H)$-Arrowing is equivalent to $H$-free Matching Removal. We show that $(P_3, H)$-Arrowing is coNP-complete for all $2$-connected $H$ except when $H = K_3$, in which case the problem is in P. We introduce a new graph invariant to help us carefully combine graphs when constructing the gadgets for our reductions. Moreover, we show how $(P_3,H)$-Arrowing hardness results can be extended to other $(F,H)$-Arrowing problems. This allows for more intuitive and palatable hardness proofs instead of ad-hoc constructions of SAT gadgets, bringing us closer to categorizing the complexity of all $(F, H)$-Arrowing problems.

Figures (14)

• Figure 1: Graphs with different $\mathrm{mepl}(G)$ values. Bold edges have $\mathrm{epl}_G = \mathrm{mepl}(G)$.
• Figure 2: Proof for Lemma \ref{['lem:mlep2-combine']} when $|V(H)| = 4$. It is easy to see that $A_{H,e}$ for $H \in \{C_4, K_4\}$ has exactly two copies of $H$ for arbitrary $e$. Moreover, constructing $A_{J_4, e}$ introduces a new $J_4$ (highlighted in red) for both nonisomorphic choices of $e \in E(J_4)$. Identified edges are bolded.
• Figure 3: (a) The graph $A'$ when $u$ and $w$ are adjacent to a red edge. (b) The graph $A'$ when either $u$ or $w$ is adjacent to a red edge. (c) The graph $B$. (d) A zoomed in look at $B$.
• Figure 4: (a) The $(P_3, H)$-signal extender described in Lemma \ref{['lem:sig-exst']}, where enforcers are labeled $EN$ and the copy of $H$ is labeled $U$. Edges whose colors are fixed in all $(P_3, H)$-good colorings have been pre-colored. The edge $(u_1,u_3)$ is dashed to signify it may or may not exist in the construction. (b) At the top, we show how extenders can be connected sequentially to form arbitrarily large extenders. The enforcers have been removed from the illustration for clarity. The in- and out-vertices are marked $a$ and $b$, respectively. At the bottom, we show how signal extenders will be depicted in our figures, where $\ell$ is the number of concatenated constructed extenders. (c) At the top, we show the coloring of the signal extender when vertex $a$ is a free vertex. At the bottom, we show the corresponding coloring of our representation of signal extenders.
• Figure 5: On the left, we show the variable gadget constructed using two copies of $A$ as described in Theorem \ref{['thm:p3-hard-1']}'s proof and signal extenders. The vertices in the square are the vertices of $A$ which had enforcers appended to them. The edge $(b,c)$ is dashed to signify that it may or may not exist. Edges have been precolored wherever possible. Note that if $(b,c)$ exists, it must be blue: if $(b,c)$ is red, the attached signal extenders will force $(a',b')$, $(b',c')$, and $(c',d')$ to be blue, forming a blue $H$. By symmetry, $(b',c')$ must also be blue. Now, observe that at least one edge in $\{(a,b), (c,d)\}$ must be red, otherwise we form a blue $H$ in $A$. Suppose $(a,b)$ is red: the signal extender forces $(a',b')$ to be blue. To avoid a blue $H$, $(c',d')$ must be red, which forces $(c,d)$ to be blue. In this case, the vertices marked $\mathbf{U}$ are nonfree vertices, and the vertices marked $\mathbf{N}$ are free. A similar pattern can be observed when we color $(c,d)$ red instead, giving us colorings where vertices marked $\mathbf{U}$ are free and vertices marked $\mathbf{N}$ are nonfree.

Theorems & Definitions (21)

• Theorem 1
• Definition 2
• Lemma 3
• Theorem 4
• Definition 5
• Definition 6: Scha
• Lemma 7
• Definition 8
• Lemma 9
• Definition 10