Further results on Hendry's Conjecture
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Statement
Suppose G is a Hamiltonian chordal graph. Is G cycle extendable if min{Δ(T) : (T, T) is a tree decomposition for G} = 4?
Context
Candidate 3 of the open problems stated in "Further results on Hendry's Conjecture", extracted for the Scalable Mathematical Discovery run.
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NEW
Problem: Reconstructed statement: for finite simple graphs, let
where is the intersection graph of connected subtrees of a tree . Question 3.8 asks whether every Hamiltonian chordal graph with is cycle extendable.
Result: The literal statement is vacuously true. In fact, no chordal graph has .
Proof. By Gavril’s characterization, every chordal graph has a representation as an intersection graph of subtrees of some tree . Given any such representation, replace every vertex of degree by a tree of maximum degree at most with distinguished leaves, one for each edge incident with . Reconnect the old incident edges through the corresponding distinguished leaves. This produces a new tree with .
For each original subtree , replace it by
together with the reconnecting edges corresponding to old edges of . Then is connected. Moreover, two original subtrees intersect iff their replacements intersect: a common old vertex becomes the common block , while disjoint old vertex sets give disjoint unions of blocks. Hence the same graph is represented on .
Thus every chordal graph satisfies . Therefore the hypothesis is impossible, including for Hamiltonian chordal graphs. The question as written has an empty hypothesis; a non-vacuous version would need an additional restriction such as using only reduced clique trees.
Citation: Gavril, F. “The intersection graphs of subtrees in trees are exactly the chordal graphs.” Journal of Combinatorial Theory, Series B 16 (1974), 47–56.
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Machine check · not human verification
machine: correctscope Full solution as submitted; SMD novelty classification TYPE1
PASS
The proof correctly attacks the literal host-tree/subtree-intersection formulation. Given any Gavril subtree representation of a chordal graph, replacing each high-degree host-tree vertex by a subcubic tree block and replacing each subtree by the union of corresponding blocks preserves exactly the intersection relation. Thus every chordal graph has such a representation with host-tree maximum degree at most 3, so the hypothesis that the minimum possible maximum degree is 4 is empty. The conclusion is therefore vacuously true, unless the intended question restricts to reduced/clique-tree decompositions, which is not stated in the supplied conjecture.
Novelty assessment
TYPE1
Classification rationale: The resolution is a routine observation: with unrestricted subtree-intersection host trees, any host tree can be made subcubic by replacing each high-degree node with a subcubic tree block and expanding each represented subtree through the block. Thus the literal hypothesis “minimum possible ” is empty. This is not a publishable standalone contribution; it is an immediate corollary of Gavril’s chordal-graph representation theorem plus a simple tree-splitting construction. If the intended question was about reduced clique trees / maximal-clique tree decompositions, this argument would not resolve that non-vacuous version.
Literature check: I checked the original paper’s Section 3.4 and searched for exact and variant formulations around Question 3.8, Hendry’s conjecture, “host tree maximum degree,” “subtrees of a binary/subcubic tree,” “intersection graphs of subtrees maximum degree 3,” and -graph terminology. I found no explicit prior statement that Question 3.8 is vacuous or that no chordal graph has this unrestricted parameter equal to 4. Related literature instead treats Gavril’s theorem, leafage, clique-tree degree, and bounded-degree host trees for path/edge-intersection classes, which are different or more restricted notions. The result therefore appears not explicitly known, but it is only a minor clarification of terminology.
Citation: Gavril, F. “The intersection graphs of subtrees in trees are exactly the chordal graphs.” Journal of Combinatorial Theory, Series B 16 (1974), 47–56.
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