Equivalent Dichotomies for Triangle Detection in Subgraph, Induced, and Colored H-Free Graphs

TL;DR

This work reduces the induced H-free case to the non-induced $\H^+$-free case, where $\H^+$ preserves the structural properties of $H$ that are relevant for the dichotomy, namely $3$-colorability and triangle count.

Abstract

A recent paper by the authors (ITCS'26) initiates the study of the Triangle Detection problem in graphs avoiding a fixed pattern $H$ as a subgraph and proposes a \emph{dichotomy hypothesis} characterizing which patterns $H$ make the Triangle Detection problem easier in $H$-free graphs than in general graphs. In this work, we demonstrate that this hypothesis is, in fact, equivalent to analogous hypotheses in two broader settings that a priori seem significantly more challenging: \emph{induced} $H$-free graphs and \emph{colored} $H$-free graphs. Our main contribution is a reduction from the induced $H$-free case to the non-induced $\H^+$-free case, where $\H^+$ preserves the structural properties of $H$ that are relevant for the dichotomy, namely $3$-colorability and triangle count. A similar reduction is given for the colored case. A key technical ingredient is a self-reduction to Unique Triangle Detection that preserves the induced $H$-freeness property, via a new color-coding-like reduction.

Figures (5)

• Figure 1: A specific $3$-coloring of $C_6$.
• Figure 2: A colored blowup of $C_9$. This graph is obtained by replacing each vertex of $C_9$ with an independent set of size $n/9$ and every edge with a complete bipartite graph. All vertices within a single blown-up set share the same color.
• Figure 3: The augmented pattern $H^{+}$ corresponding to $P_5$. The original pattern $P_5$ is drawn in black, and the additional parts are drawn in gray. Observe that if an induced $P_5$-free graph contains $H^{+}$ as a subgraph, then the vertices corresponding to $P_5$ must be connected by at least one non-edge of $P_5$, e.g., the edge $v_1v_4$. In that case, the graph contains at least two triangles: $v_1v_4x_{v_1v_4}$ and $v_1v_4y_{v_1v_4}$.
• Figure 6: The augmented pattern $P_6^+$ with a degenerate coloring. The edges of the original $P_6$ are represented by highlighted purple lines. Wedges connecting the endpoints of non-edges are shown in gray whenever the endpoints share the same color; this is because such paths do not obstruct the degenerate coloring. From this coloring, it is evident that $P_6^+$ admits a degenerate coloring. Consequently, Triangle Detection in induced $P_6$-free graphs can be solved in subcubic time using abboud2026triangle.
• Figure 7: The augmented pattern $C_7^+$ with a degenerate coloring. The edges of the original $C_7$ are represented by highlighted purple lines. Paths connecting the endpoints of non-edges are shown in gray whenever the endpoints share the same color; this is because such paths do not obstruct the degenerate coloring. (Note: For visual clarity, only one wedge is shown for these connections, rather than the two required by the formal definition of $H^{+}$). From this coloring, it is evident that $C_7^+$ admits a degenerate coloring. Consequently, Triangle Detection in induced $C_7$-free graphs can be solved in subcubic time using abboud2026triangle.

Theorems & Definitions (15)

• Conjecture 1: BMM
• theorem 1: Informal, see Theorem \ref{['thm:main']}
• Definition 1: Colored copy of $H$ in $G$
• theorem 2: Informal, see Theorem \ref{['thm:main']}
• theorem 3
• theorem 4: Color-coding induced $H$-free graphs
• theorem 5
• Definition 2
• lemma 6: Reduction to Unique Triangle Detection
• theorem 6: Color-coding induced $H$-free graphs