Hamiltonian Path Index

• Hamiltonian Path Index is defined as the minimum number of line-graph iterations required for a graph to contain a Hamiltonian path, ensuring traceability for every connected graph.
• It distinguishes itself from the classical Hamiltonian index by focusing on paths rather than cycles, with exact formulas available for trees and structural characterizations via dominating trails.
• Practical estimates and upper bounds are derived using subgraph families like EUPₖ(G) and parameters such as tree branch geometry, trail length, and graph diameter.

The Hamiltonian path index of a connected graph measures how many iterations of the line-graph operator are needed before a Hamiltonian path appears. For a graph $G$, the iterated line graphs are defined by $L^0(G)=G$, $L^1(G)=L(G)$, and $L^n(G)=L(L^{n-1}(G))$, and the invariant is

$$h_p(G)=\min\{n\ge 0: L^n(G)\text{ contains a Hamiltonian path}\}.$$

Equivalently, $h_p(G)$ is the minimum $n$ such that $L^n(G)$ is traceable. A characterization of traceable iterated line graphs established this invariant in a structural form, and a later treatment emphasized that, unlike the classical Hamiltonian index based on Hamiltonian cycles, $h_p(G)$ exists for every connected graph and admits an exact formula for trees (Nou et al., 2020, Ekstein et al., 30 Jul 2025).

1. Definition through iterated line graphs

For a graph $G$, the line graph $L^0(G)=G$0 has vertex set $L^0(G)=G$1, with two vertices adjacent in $L^0(G)=G$2 exactly when the corresponding edges of $L^0(G)=G$3 share an endpoint. The iterated line graphs are formed recursively, and the Hamiltonian path index asks for the least iteration at which traceability appears. In the terminology used in the literature,

$L^0(G)=G$4

The invariant is studied for finite connected undirected graphs. One formulation specifies that the iteration $L^0(G)=G$5 is considered as long as the previous iterate has a nonempty edge set (Ekstein et al., 30 Jul 2025). For the base case $L^0(G)=G$6, traceability of the line graph is governed by a trail condition: $L^0(G)=G$7 A dominating trail is a trail $L^0(G)=G$8 such that every edge of $L^0(G)=G$9 has at least one endpoint on $L^1(G)=L(G)$0 (Ekstein et al., 30 Jul 2025).

A basic existence fact distinguishes $L^1(G)=L(G)$1 from several older Hamiltonian invariants. If $L^1(G)=L(G)$2 is a path, then $L^1(G)=L(G)$3 itself already has a Hamiltonian path, so

$L^1(G)=L(G)$4

If $L^1(G)=L(G)$5 is not a path, then the classical Hamiltonian index $L^1(G)=L(G)$6 exists and satisfies

$L^1(G)=L(G)$7

Hence $L^1(G)=L(G)$8 exists for every connected graph (Ekstein et al., 30 Jul 2025).

2. Relation to the classical Hamiltonian index

The Hamiltonian path index is closely related to, but distinct from, the classical Hamiltonian index of iterated line graphs. The latter is defined by Hamiltonian cycles rather than Hamiltonian paths.

Invariant	Definition	Basic feature
$L^1(G)=L(G)$9	Least $L^n(G)=L(L^{n-1}(G))$0 such that $L^n(G)=L(L^{n-1}(G))$1 has a Hamiltonian path	Exists for every connected graph
$L^n(G)=L(L^{n-1}(G))$2	Least $L^n(G)=L(L^{n-1}(G))$3 such that $L^n(G)=L(L^{n-1}(G))$4 has a Hamiltonian cycle	Classical cycle-based invariant

This distinction is substantive. The 2019 algorithmic work on Hamiltonian index explicitly studies

$L^n(G)=L(L^{n-1}(G))$5

and states that it does not define or study a separate Hamiltonian path index (Philip et al., 2019). In that cycle-based theory, the relevant one-step criterion is

$L^n(G)=L(L^{n-1}(G))$6

whereas the path-based theory replaces dominating closed trails by dominating trails (Philip et al., 2019).

The difference is especially visible on paths. In the path-based setting,

$L^n(G)=L(L^{n-1}(G))$7

because a path is already traceable. By contrast, the classical Hamiltonian index behaves differently on paths, since iterated line graphs of a path remain paths and never become Hamiltonian cycles in the usual sense (Ekstein et al., 30 Jul 2025).

A common misconception is therefore to treat “Hamiltonian path index” as merely a minor variant of $L^n(G)=L(L^{n-1}(G))$8. The literature instead treats it as a separate invariant with its own structural characterization, its own extremal behavior on trees, and a different relationship to block structure (Nou et al., 2020, Ekstein et al., 30 Jul 2025).

3. Structural characterization of traceable iterated line graphs

The principal general characterization uses a family of subgraphs $L^n(G)=L(L^{n-1}(G))$9, introduced as the path-analogue of the $$h_p(G)=\min\{n\ge 0: L^n(G)\text{ contains a Hamiltonian path}\}.$$0 structures used earlier for Hamiltonian cycles in iterated line graphs (Nou et al., 2020).

Several graph-theoretic notions enter the definition. For a subgraph $$h_p(G)=\min\{n\ge 0: L^n(G)\text{ contains a Hamiltonian path}\}.$$1 of $$h_p(G)=\min\{n\ge 0: L^n(G)\text{ contains a Hamiltonian path}\}.$$2,

$$h_p(G)=\min\{n\ge 0: L^n(G)\text{ contains a Hamiltonian path}\}.$$3

and for a graph $$h_p(G)=\min\{n\ge 0: L^n(G)\text{ contains a Hamiltonian path}\}.$$4,

$$h_p(G)=\min\{n\ge 0: L^n(G)\text{ contains a Hamiltonian path}\}.$$5

A branch in $$h_p(G)=\min\{n\ge 0: L^n(G)\text{ contains a Hamiltonian path}\}.$$6 is a nontrivial path with ends in $$h_p(G)=\min\{n\ge 0: L^n(G)\text{ contains a Hamiltonian path}\}.$$7 and internal vertices, if any, of degree $$h_p(G)=\min\{n\ge 0: L^n(G)\text{ contains a Hamiltonian path}\}.$$8. Let $$h_p(G)=\min\{n\ge 0: L^n(G)\text{ contains a Hamiltonian path}\}.$$9 be the set of branches, and

$h_p(G)$0

For a positive integer $h_p(G)$1, $h_p(G)$2 is the set of subgraphs $h_p(G)$3 of $h_p(G)$4 satisfying:

1. $h_p(G)$5;
2. $h_p(G)$6;
3. $h_p(G)$7 for every subgraph $h_p(G)$8 of $h_p(G)$9;
4. $n$0 for every branch $n$1 with $n$2;
5. $n$3 for every branch $n$4 with $n$5.

The central theorem is then: $n$6 for connected graphs $n$7 with at least three edges and $n$8 (Nou et al., 2020).

Two supporting results organize the proof. First, the line-graph step itself preserves the characterization: $n$9 Second, the base case is

$L^n(G)$0

The proof strategy parallels the Hamiltonian-cycle theory, but weakens the even-degree condition from $L^n(G)$1 to $L^n(G)$2, reflecting the passage from Eulerian closed structures to trail structures (Nou et al., 2020).

The scope restriction $L^n(G)$3 is essential. The theorem does not hold for $L^n(G)$4; the one-step case is handled instead by the dominating-trail characterization of $L^n(G)$5 (Nou et al., 2020).

4. Exact determination for trees

The most explicit closed formula currently available is for trees. A 2025 treatment gives an exact expression for $L^n(G)$6 for every tree $L^n(G)$7 (Ekstein et al., 30 Jul 2025).

A branch in a graph is a nontrivial path whose end vertices have degree different from $L^n(G)$8, and whose internal vertices, if any, all have degree $L^n(G)$9. For a tree $h_p(G)$0, the relevant branch families are:

• $h_p(G)$1: all branches;
• $h_p(G)$2: branches all of whose edges are bridges;
• $h_p(G)$3: branches in $h_p(G)$4 with an end vertex of degree $h_p(G)$5.

For each branch $h_p(G)$6 of a tree $h_p(G)$7, define

$h_p(G)$8

Thus leaf-branches contribute their length, while internal bridge-branches contribute one more than their length.

If $h_p(G)$9 is not a path, choose branches $G$0 such that $G$1 is maximal among all pairs of branches. Let $G$2 be the set of all endpaths containing $G$3 and $G$4, and for $G$5, let $G$6 denote all branches of $G$7 contained in $G$8. The main theorem is

$G$9

and otherwise

$L^0(G)=G$00

The formula identifies the obstruction precisely: one chooses a principal endpath containing two maximal branches, and the Hamiltonian path index is the least possible worst $L^0(G)=G$01-value among branches lying outside that endpath.

Several special cases are immediate (Ekstein et al., 30 Jul 2025). If $L^0(G)=G$02 with $L^0(G)=G$03, every branch is a one-edge leaf-branch, so $L^0(G)=G$04 for all branches and

$L^0(G)=G$05

More generally, every non-path caterpillar satisfies

$L^0(G)=G$06

A symmetric $L^0(G)=G$07-shaped tree with three arms of length $L^0(G)=G$08 has three leaf-branches with $L^0(G)=G$09; any endpath uses two of them, leaving one off-path branch of value $L^0(G)=G$10, hence

$L^0(G)=G$11

The proof proceeds by analyzing how branches shorten under repeated line-graph iteration. Leaf-branches disappear after $L^0(G)=G$12 steps once $L^0(G)=G$13, while internal bridge-branches persist one step longer, which explains the $L^0(G)=G$14 in their definition (Ekstein et al., 30 Jul 2025).

5. Upper bounds and computable estimates

The structural characterization via $L^0(G)=G$15 yields several explicit upper bounds for the Hamiltonian path index (Nou et al., 2020). These bounds are designed to be checked directly from large-scale graph parameters such as maximum trail length, diameter, and degree.

Let $L^0(G)=G$16 be a trail with the maximum number of vertices, and among those, one minimizing the number of vertices of degree at least $L^0(G)=G$17 left outside the trail. Define

$L^0(G)=G$18

and

$L^0(G)=G$19

Then

$L^0(G)=G$20

A direct corollary is

$L^0(G)=G$21

and since $L^0(G)=G$22,

$L^0(G)=G$23

A degree-based estimate uses

$L^0(G)=G$24

together with

$L^0(G)=G$25

Then

$L^0(G)=G$26

For connected simple graphs this yields the simpler bound

$L^0(G)=G$27

These estimates are significant because they convert a definition based on iterated line graphs into inequalities involving ordinary graph parameters. In the source presentation, the bounds derived from $L^0(G)=G$28, $L^0(G)=G$29, and $L^0(G)=G$30 are presented as applications of the traceability characterization, and several of them are stated to be sharp (Nou et al., 2020).

6. Scope, limitations, and current structure of the theory

The present theory has a clear asymmetry between general graphs and trees. On the general side, the characterization

$L^0(G)=G$31

for $L^0(G)=G$32 gives a complete criterion in structural terms, but it is not a closed formula comparable to the tree theorem (Nou et al., 2020). On the tree side, the invariant is completely determined by branch geometry (Ekstein et al., 30 Jul 2025).

A second limitation is that analogies with the classical Hamiltonian index do not transfer wholesale. The 2025 treatment states that Saražin proved an equality phenomenon for the cycle-based Hamiltonian index on trees and graphs with Hamiltonian $L^0(G)=G$33-blocks, but that “this is not true for Hamiltonian path index” (Ekstein et al., 30 Jul 2025). The abstract of that work says it discusses graphs with Hamiltonian $L^0(G)=G$34-connected blocks, yet no full general theorem for that class is stated in the provided account.

A third point of care is terminological. The line-graph Hamiltonian index studied in algorithmic and parameterized form in the cycle literature is a different invariant. That work concerns the least $L^0(G)=G$35 for which $L^0(G)=G$36 has a Hamiltonian cycle, not a Hamiltonian path (Philip et al., 2019). The two theories share the same line-graph framework and some of the same trail-based ideas, but they diverge at the level of both definition and structure.

The contemporary understanding of Hamiltonian path index can therefore be summarized as follows. It is a path-based line-graph iteration invariant, always defined on connected graphs, controlled at one step by dominating trails, characterized for $L^0(G)=G$37 by the subgraph family $L^0(G)=G$38, and completely solved for trees through an exact branch formula (Nou et al., 2020, Ekstein et al., 30 Jul 2025). Beyond trees, the invariant is structurally understood but not yet reduced to comparably explicit formulas.