Uniquely $K_r$-Saturated Graphs

Hartke, Stephen G.; Stolee, Derrick

doi:10.37236/2162

Cited by 5 publications

(8 citation statements)

References 18 publications

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“…A recent development is the discovery that certain circulant graphs G(n, S) are uniquely K rsaturated, including three infinite families [21]. The first step in proving this property is showing that the clique number of G(n, S) is equal to r − 1.…”

Section: Introductionmentioning

confidence: 99%

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On the independence ratio of distance graphs

Carraher¹,

Galvin²,

Hartke³

et al. 2016

Discrete Mathematics

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A distance graph is an undirected graph on the integers where two integers are adjacent if their difference is in a prescribed distance set. The independence ratio of a distance graph G is the maximum density of an independent set in G. Lih, Liu, and Zhu [31] showed that the independence ratio is equal to the inverse of the fractional chromatic number, thus relating the concept to the well studied question of finding the chromatic number of distance graphs.We prove that the independence ratio of a distance graph is achieved by a periodic set, and we present a framework for discharging arguments to demonstrate upper bounds on the independence ratio. With these tools, we determine the exact independence ratio for several infinite families of distance sets of size three, determine asymptotic values for others, and present several conjectures.

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Section: Introductionmentioning

confidence: 99%

“…In this complement, the independence number is of particular interest. Our discharging method is an adaptation of the discharging method used in [21] to determine the independence number in circulant graphs.…”

Section: Introductionmentioning

confidence: 99%

On the independence ratio of distance graphs

Carraher¹,

Galvin²,

Hartke³

et al. 2016

Discrete Mathematics

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“…The Petersen graph, a uniquely K 3 -saturated Moore graph, corresponds to the entry (n = r + 7, r = 3), and the entries n = r + 6 and n = r + 8 in the column r = 4 correspond to two sporadic uniquely K 4 -saturated graphs discovered by Cooper and Collins. The r = 4 and r = 6 entries in the n = r + 9 row correspond to uniquely K r -saturated graphs discovered by Hartke and Stolee [8]. Other than these, few primitive uniquely K r -saturated graphs exist on n vertices for small values of n. We were not able to solve our integer program for larger values of n and r, so Table 1 includes the results of the computational study included in [8].…”

Section: Computational Searchmentioning

confidence: 99%

“…The study of H-saturated graphs most relevant to this paper was undertaken by Hartke and Stolee [8], who examined the case H = K r . The uniquely K 3 -saturated graphs were already characterized in [2] since K 3 = C 3 , but prior to this study, few examples of these graphs were known for r > 3.…”

Section: Introductionmentioning

confidence: 99%

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Uniquely $K_r^{(k)}$-Saturated Hypergraphs

Gyárfás

Hartke

Viss

2018

Electron. J. Combin.

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In this paper we generalize the concept of uniquely $K_r$-saturated graphs to hypergraphs. Let $K_r^{(k)}$ denote the complete $k$-uniform hypergraph on $r$ vertices. For integers $k,r,n$ such that $2\leqslant k <r<n$, a $k$-uniform hypergraph $H$ with $n$ vertices is uniquely $K_r^{(k)}$-saturated if $H$ does not contain $K_r^{(k)}$ but adding to $H$ any $k$-set that is not a hyperedge of $H$ results in exactly one copy of $K_r^{(k)}$. Among uniquely $K_r^{(k)}$-saturated hypergraphs, the interesting ones are the primitive ones that do not have a dominating vertex—a vertex belonging to all possible ${n-1\choose k-1}$ edges. Translating the concept to the complements of these hypergraphs, we obtain a natural restriction of $\tau$-critical hypergraphs: a hypergraph $H$ is uniquely $\tau$-critical if for every edge $e$, $\tau(H-e)=\tau(H)-1$ and $H-e$ has a unique transversal of size $\tau(H)-1$.We have two constructions for primitive uniquely $K_r^{(k)}$-saturated hypergraphs. One shows that for $k$ and $r$ where $4\leqslant k<r\leqslant 2k-3$, there exists such a hypergraph for every $n>r$. This is in contrast to the case $k=2$ and $r=3$ where only the Moore graphs of diameter two have this property. Our other construction keeps $n-r$ fixed; in this case we show that for any fixed $k\ge 2$ there can only be finitely many examples. We give a range for $n$ where these hypergraphs exist. For $n-r=1$ the range is completely determined: $k+1\leqslant n \leqslant {(k+2)^2\over 4}$. For larger values of $n-r$ the upper end of our range reaches approximately half of its upper bound. The lower end depends on the chromatic number of certain Johnson graphs.

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Uniquely Cycle‐Saturated Graphs

Wenger

West

2016

Journal of Graph Theory

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Given a graph F , a graph G is uniquely F -saturated if F is not a subgraph of G and adding any edge of the complement to G completes exactly one copy of F . In this paper we study uniquely C t -saturated graphs. We prove the following: (1) a graph is uniquely C 5 -saturated if and only if it is a friendship graph. (2) There are no uniquely C 6 -saturated graphs or uniquely C 7 -saturated graphs. (3) For t ≥ 6, there are only finitely many uniquely C t -saturated graphs (we conjecture that in fact there are none).

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Uniquely $K_r$-Saturated Graphs

Cited by 5 publications

References 18 publications

On the independence ratio of distance graphs

On the independence ratio of distance graphs

Uniquely $K_r^{(k)}$-Saturated Hypergraphs

Uniquely Cycle‐Saturated Graphs

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