Strong Central 2-Trees with Tail Degrees {2, 3}: Structural Characterization and Uniqueness Criteria
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| Format: | Preprint |
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2025
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| author | Allagan, Julian Langley, Shawn Gao, Weizheng Elbakary, Mohamed |
| author_facet | Allagan, Julian Langley, Shawn Gao, Weizheng Elbakary, Mohamed |
| contents | We study strong $r$-central $2$-trees whose non-central vertices have degrees in $\{2,3\}$, focusing on the cases $r=1,2,3$. For each $r$, we derive exact degree constraints relating the maximum degree $Δ$ to the numbers of degree-$3$ and degree-$2$ tail vertices. In the unicentral case ($r=1$), we prove that the fan graph is the unique realization for all $n\ge 3$. For bicentral $2$-trees ($r=2$), we show that the number of degree-$3$ vertices is always even, establish sharp uniqueness results for $x\in\{0,2\}$, prove existence for all feasible values of $Δ$, and obtain linear lower bounds on the number of non-isomorphic realizations. For tricentral $2$-trees ($r=3$), we characterize extremal configurations, establish a divisibility constraint on the tail parameters, and prove a quadratic lower bound on the number of non-isomorphic graphs for infinitely many values of $n$. These results provide a unified structural framework for central $2$-trees with bounded tail degrees and highlight sharp transitions between rigidity and combinatorial growth. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_18378 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strong Central 2-Trees with Tail Degrees {2, 3}: Structural Characterization and Uniqueness Criteria Allagan, Julian Langley, Shawn Gao, Weizheng Elbakary, Mohamed Combinatorics 05C05, 05C07, 05C30, 05C60 We study strong $r$-central $2$-trees whose non-central vertices have degrees in $\{2,3\}$, focusing on the cases $r=1,2,3$. For each $r$, we derive exact degree constraints relating the maximum degree $Δ$ to the numbers of degree-$3$ and degree-$2$ tail vertices. In the unicentral case ($r=1$), we prove that the fan graph is the unique realization for all $n\ge 3$. For bicentral $2$-trees ($r=2$), we show that the number of degree-$3$ vertices is always even, establish sharp uniqueness results for $x\in\{0,2\}$, prove existence for all feasible values of $Δ$, and obtain linear lower bounds on the number of non-isomorphic realizations. For tricentral $2$-trees ($r=3$), we characterize extremal configurations, establish a divisibility constraint on the tail parameters, and prove a quadratic lower bound on the number of non-isomorphic graphs for infinitely many values of $n$. These results provide a unified structural framework for central $2$-trees with bounded tail degrees and highlight sharp transitions between rigidity and combinatorial growth. |
| title | Strong Central 2-Trees with Tail Degrees {2, 3}: Structural Characterization and Uniqueness Criteria |
| topic | Combinatorics 05C05, 05C07, 05C30, 05C60 |
| url | https://arxiv.org/abs/2512.18378 |