Connected coverings and an application to oriented matroids

Forge, David; Alfonsín, Jorge Luis Ramírez

doi:10.1016/s0012-365x(97)00246-x

Cited by 3 publications

(27 citation statements)

References 7 publications

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“…Forge and Ramírez Alfonsín proved that

CC (n, r) \geq \frac{(\frac{0pt}{n}) - 1}{r} = : {CC}_{1}^{*} (n, r) .

Moreover, Sidorenko proved that

T (n, r + 1, r) \geq (\frac{n - r}{n - r + 1}) \frac{((), \binom{n}{r})}{r}

. Together with , we obtain that

\begin{matrix} true \\ right CC (n, r) \geq normalC (n, r) = normalT (n, n - r, n - r - 1) \geq ((), \frac{r + 1}{r + 2}) \frac{(\frac{0pt}{n})}{n - r - 1} = : {CC}_{2}^{*} (n, r) . \end{matrix}

Combining and , together with a straight forward computation we have

CC (n, r) \geq max {{CC}_{1}^{*} (n, r), {CC}_{2}^{*} (n, r)},

where the maximum is attained by the second term if and only if

r \geq \frac{2}{3} (n - 1)

.…”

Section: Basic Resultsmentioning

confidence: 73%

“…The study of

C (n, r)

and

CC (n, r)

seems to be interesting not only for Design Theory but also, in view of Equations and , for the implications on the behavior of

s (n, r)

in Oriented Matroid Theory. This relationship was already remarked in [, Theorem ] where it was proved that

CC (n, r) \leq 2 C (n, r) .

The latter can be slightly improved as follows

CC (n, r) \leq 2 C (n, r) - 1,

since the graph G associated to a covering with

C (n, r)

blocks (and thus with

| V (G) | = C (n, r)

) can be made connected by adding at most

C (n, r) - 1

extra vertices (blocks), obtaining a graph corresponding to a

(n, r + 1, r)

‐connected covering with at most

2 C (n, r) - 1

blocks.…”

Section: Introductionmentioning

confidence: 67%

“…Bounds and precise values for all

CC (n, r)

with

n \leq 14

are given in Table . All the exact values previously given in for the same range have been improved by using our above results.…”

Section: Results For Small and Large Rmentioning

confidence: 90%

“…Forge and Ramírez Alfonsín [, Theorem 4.2] proved that

CC (n, r) \leq \sum_{i = r + 1}^{n - 1} C (i, r - 1)

. In this proof, they used the following recursive upper bound that will be useful for us later,

CC (n, r) \leq CC (n - 1, r) + C (n - 1, r - 1) .

…”

Section: Basic Resultsmentioning

confidence: 99%

“…In , Forge and Ramírez Alfonsín introduced the notion of connected coverings and proved that

s (n, r) \leq CC (n, r) .

The latter was then used to improve the best upper bound,

s (n, r) \leq (\frac{0pt}{n - 1})

, known at that time due to Hamidoune and Las Vergnas ; see also for related results.…”

Section: Introductionmentioning

confidence: 99%

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Connected Covering Numbers

et al. 2015

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A connected covering is a design system in which the corresponding block graph is connected. The minimum size of such coverings are called connected coverings numbers. In this paper, we present various formulas and bounds for several parameter settings for these numbers. We also investigate results in connection with Turán systems. Finally, a new general upper bound, improving an earlier result, is given. The latter is used to improve upper bounds on a question concerning oriented matroid due to Las Vergnas.

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“…Forge and Ramírez Alfonsín proved that

CC (n, r) \geq \frac{(\frac{0pt}{n}) - 1}{r} = : {CC}_{1}^{*} (n, r) .

Moreover, Sidorenko proved that

T (n, r + 1, r) \geq (\frac{n - r}{n - r + 1}) \frac{((), \binom{n}{r})}{r}

. Together with , we obtain that

\begin{matrix} true \\ right CC (n, r) \geq normalC (n, r) = normalT (n, n - r, n - r - 1) \geq ((), \frac{r + 1}{r + 2}) \frac{(\frac{0pt}{n})}{n - r - 1} = : {CC}_{2}^{*} (n, r) . \end{matrix}

Combining and , together with a straight forward computation we have

CC (n, r) \geq max {{CC}_{1}^{*} (n, r), {CC}_{2}^{*} (n, r)},

where the maximum is attained by the second term if and only if

r \geq \frac{2}{3} (n - 1)

.…”

Section: Basic Resultsmentioning

confidence: 73%

“…The study of

C (n, r)

and

CC (n, r)

seems to be interesting not only for Design Theory but also, in view of Equations and , for the implications on the behavior of

s (n, r)

in Oriented Matroid Theory. This relationship was already remarked in [, Theorem ] where it was proved that

CC (n, r) \leq 2 C (n, r) .

The latter can be slightly improved as follows

CC (n, r) \leq 2 C (n, r) - 1,

since the graph G associated to a covering with

C (n, r)

blocks (and thus with

| V (G) | = C (n, r)

) can be made connected by adding at most

C (n, r) - 1

extra vertices (blocks), obtaining a graph corresponding to a

(n, r + 1, r)

‐connected covering with at most

2 C (n, r) - 1

blocks.…”

Section: Introductionmentioning

confidence: 67%

“…Bounds and precise values for all

CC (n, r)

with

n \leq 14

are given in Table . All the exact values previously given in for the same range have been improved by using our above results.…”

Section: Results For Small and Large Rmentioning

confidence: 90%

“…Forge and Ramírez Alfonsín [, Theorem 4.2] proved that

CC (n, r) \leq \sum_{i = r + 1}^{n - 1} C (i, r - 1)

. In this proof, they used the following recursive upper bound that will be useful for us later,

CC (n, r) \leq CC (n - 1, r) + C (n - 1, r - 1) .

…”

Section: Basic Resultsmentioning

confidence: 99%

“…In , Forge and Ramírez Alfonsín introduced the notion of connected coverings and proved that

s (n, r) \leq CC (n, r) .

The latter was then used to improve the best upper bound,

s (n, r) \leq (\frac{0pt}{n - 1})

, known at that time due to Hamidoune and Las Vergnas ; see also for related results.…”

Section: Introductionmentioning

confidence: 99%

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Connected Covering Numbers

et al. 2015

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How Many Circuits Determine an Oriented Matroid?

2017

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Las Vergnas & Hamidoune studied the number of circuits needed to determine an oriented matroid. In this paper we investigate this problem and some new variants, as well as their interpretation in particular classes of matroids. We present general upper and lower bounds in the setting of general connected orientable matroids, leading to the study of subgraphs of the base graph and the intersection graph of circuits.We then consider the problem for uniform matroids which is closely related to the notion of (connected) covering numbers in Design Theory. Finally, we also devote special attention to regular matroids as well as some graphic and cographic matroids leading in particular to the topics of (connected) bond and cycle covers in Graph Theory.

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