Tough Coline Graphs: Hamiltonicity and Traceability

• Tough coline graphs are defined as the complements of line graphs of 1-tough graphs, where vertices represent non-incident edges of the original graph.
• Mammoliti’s classification shows that apart from four specific exceptions (like K5 and related graphs), these graphs guarantee Hamiltonicity through robust structural properties.
• The study extends to traceability using pseudo-toughness and longest-cycle techniques, establishing strong links with cyclic matching-sequenceability and graph connectivity.

A tough coline graph is the complement of a line graph of a simple graph $G$ that is $1$-tough—that is, for every vertex cut $S$, the number of connected components after deleting $S$ is at most $|S|$. Coline graphs (denoted $\co(G)$) have as vertices the edges of $G$; two vertices in $\co(G)$ are adjacent precisely when their corresponding edges in $G$ are non-incident. The structural and Hamiltonian properties of these graphs, particularly under the assumption of toughness, have been the subject of several characterizations culminating in a definitive classification for Hamiltonicity and traceability as provided by Mammoliti (Mammoliti, 31 Jan 2026).

1. Definitions and Fundamental Constructions

Let $G=(V,E)$ be a simple graph. The line graph $L(G)$ is defined on vertex set $E$ where two vertices (edges of $G$) are adjacent if and only if they share a common endpoint in $G$. The coline graph $\co(G)$ is the complement of this line graph on $E$: for $e, e' \in E(G)$,

$ee' \in E(\co(G)) \quad \Longleftrightarrow \quad e, e' \text{ are non-incident in } G.$

A graph $H$ is $t$-tough if for every vertex subset $S \subseteq V(H)$, $c(H-S) \leq |S|/t$, where $c(H-S)$ is the number of components of $H-S$. The focus here is on $1$-toughness.

2. Hamiltonicity in Tough Coline Graphs

A principal result (“Mammoliti’s theorem”) provides a near-complete classification of when a tough coline graph is Hamiltonian. If $\co(G)$ is tough, then $\co(G)$ is Hamiltonian unless $G$ is isomorphic to one of four exceptional graphs: $K_5, \quad H_1, \quad H_2, \quad H_3,$ where:

• $K_5$ is the complete graph on five vertices,
• $H_1$ is the triangle with one leaf attached to each vertex ($K_3 \circ K_1$),
• $H_2$ is $H_1$ with one leaf-edge subdivided once,
• $H_3$ is $H_1$ with two leaf-edges each subdivided once.

In all four cases, $\co(G)$ remains $1$-tough but fails to admit a Hamiltonian cycle. For all other root graphs $G$, toughness of $\co(G)$ guarantees Hamiltonicity. Notably, $\co(K_5)$ is the Petersen graph, which is $1$-tough but non-Hamiltonian (Mammoliti, 31 Jan 2026).

3. Traceability and Pseudo-Toughness

Hamiltonian path analogues are captured through the concept of traceability. For this, pseudo-toughness is defined: a graph $L$ is pseudo-tough if the graph $L^*$, formed by adjoining one new universal vertex to $L$, is $1$-tough.

A characterization arises:

• $\co(G)$ is traceable if and only if $\co(G)$ is pseudo-tough and $G$ is not $K_3 \circ K_1$.
• In the lone exception, $\co(K_3 \circ K_1)$ fails to contain a Hamiltonian path, though it is pseudo-tough.

This result leverages the equivalence between Hamiltonicity in the augmented graph $L^*$ and traceability in $L$ via a dominating vertex construction, and follows from the main Hamiltonicity theorem (Mammoliti, 31 Jan 2026).

4. Longest-Cycle Proof Techniques and Key Lemmas

The proof that toughness enforces Hamiltonicity except for four roots is based on a refined longest-cycle method. If $L = \co(G)$ is tough but non-Hamiltonian, and $C$ is a longest cycle in $L$, several structural phenomena arise:

• $L$ is $2$-connected, so every component of $L - V(C)$ is a singleton.
• For $x \notin V(C)$ and its neighbors $N_L(x) = \{x_1, \ldots, x_d\} \subseteq V(C)$ in $C$'s cyclic order, with $x_i^+, x_i^-$ denoting successor/predecessor, the following hold:

Key Lemmas

Lemma	Description	Structural Implication
Neighbourhood-Separation (4.1)	For each neighbor $u$ of $x$, none of $u^+, u^-$ are also neighbors; there are no paths with interior in $L-V(C)$ connecting $u^+, v^+$; certain vertex sets are independent	Restricts how external vertices may connect to $C$
Four-Chord-Forbidden (4.2)	For any three consecutive neighbors, either all four specified chords are non-edges, or one particular non-edge is present	Precludes chord configurations that allow cycle extension
Edge-Non-Extendability (4.3)	For any chord $ww^+$ between two external neighbor indices, at least one of two "corner" edges is missing	Prevents extension via new cycle construction

These lemmas lead to a constrained case analysis: only in the four exceptional root graphs $G$ does the structure evade Hamiltonicity. In all other cases, a contradiction to cycle maximality arises (Mammoliti, 31 Jan 2026).

5. Structural Corollaries and Further Characterizations

Mammoliti synthesizes earlier results of Wu–Meng and Liu, identifying the root graphs of non-tough coline graphs as precisely those with too few edges, certain adjacency structures in vertices of maximum degree, or matching a short catalog of small graphs (see Corollary 4.2 in (Mammoliti, 31 Jan 2026)). The same framework, lowering the edge count by one, describes all $\co(G)$ failing traceability.

An application arises for cyclic matching-sequenceability: $\cms(G) \geq 2$ if and only if $\co(G)$ is Hamiltonian. Therefore, the Hamiltonicity theorem fully classifies graphs $G$ by their cyclic matching-sequenceability of degree two or higher.

6. Significance and Connections

The compactness of Mammoliti’s longest-cycle argument provides both an alternative and a sharpening of established Hamiltonicity and traceability criteria for the complements of line graphs, clarifying the role of toughness in enforcing cycle and path properties. The isolation of a minimal set of exceptional root graphs underscores the strength of toughness in this structural context and establishes deeper links to matching theory and componentwise connectivity in combinatorial optimization (Mammoliti, 31 Jan 2026).