On diregular digraphs with degree two and excess three

Tuite, James

doi:10.1016/j.dam.2019.02.045

Cited by 3 publications

(2 citation statements)

References 26 publications

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“…Further results on the problem and examples of geodetic cages can be found in [25,26,27,28]. As no non-trivial digraphs with excess one have been found, we make the following conjecture.…”

Section: Introductionmentioning

confidence: 70%

The structure of digraphs with excess one

Tuite¹

2021

Preprint

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A digraph G is k-geodetic if for any (not necessarily distinct) vertices u, v there is at most one directed walk from u to v with length not exceeding k. The order of a k-geodetic digraph with minimum out-degree d is bounded below by the directed Moore boundThe Moore bound can be met only in the trivial cases d = 1 and k = 1, so it is of interest to look for k-geodetic digraphs with out-degree d and smallest possible order M (d, k) + ǫ, where ǫ is the excess of the digraph. Miller, Miret and Sillasen recently ruled out the existence of digraphs with excess one for k = 3, 4 and d ≥ 2 and for k = 2 and d ≥ 8. We conjecture that there are no digraphs with excess one for d, k ≥ 2 and in this paper we investigate the structure of minimal counterexamples to this conjecture. We severely constrain the possible structures of the outlier function and prove the non-existence of certain digraphs with degree three and excess one, as well closing the open cases k = 2 and d = 3, 4, 5, 6, 7 left by the analysis of Miller et al. We further show that there are no involutary digraphs with excess one, i.e. the outlier function of any such digraph must contain a cycle of length ≥ 3.

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“…Further results on the problem and examples of geodetic cages can be found in [25,26,27,28]. As no non-trivial digraphs with excess one have been found, we make the following conjecture.…”

Section: Introductionmentioning

confidence: 70%

The structure of digraphs with excess one

Tuite¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Further results on the problem and examples of geodetic cages can be found in [23][24][25][26]. As no nontrivial digraphs with excess one have been found, we make the following conjecture.…”

Section: Introductionmentioning

confidence: 72%

The structure of digraphs with excess one

Tuite

2024

Journal of Graph Theory

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A digraph is ‐geodetic if for any (not necessarily distinct) vertices there is at most one directed walk from to with length not exceeding . The order of a ‐geodetic digraph with minimum out‐degree is bounded below by the directed Moore bound . The Moore bound can be met only in the trivial cases and , so it is of interest to look for ‐geodetic digraphs with out‐degree and smallest possible order , where is the excess of the digraph. Miller, Miret and Sillasen recently ruled out the existence of digraphs with excess one for and and for and . We conjecture that there are no digraphs with excess one for and in this paper we investigate the structure of minimal counterexamples to this conjecture. We severely constrain the possible structures of the outlier function and prove the nonexistence of certain digraphs with degree three and excess one, as well closing the open cases and left by the analysis of Miller et al. We further show that there are no involutary digraphs with excess one, that is, the outlier function of any such digraph must contain a cycle of length .

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New Bounds on k-Geodetic Digraphs and Mixed Graphs

Tuite

Erskine

2021

Trends in Mathematics

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On diregular digraphs with degree two and excess three

Cited by 3 publications

References 26 publications

The structure of digraphs with excess one

The structure of digraphs with excess one

The structure of digraphs with excess one

New Bounds on k-Geodetic Digraphs and Mixed Graphs

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