Distance edge-colourings and matchings

Kang, Ross J.; Manggala, Putra

doi:10.1016/j.dam.2012.07.001

Cited by 6 publications

(5 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Kang and Manggala [5] showed an upper bound of um k (G) when the expected degree d = np is a large enough constant c, which improves the result in [3]. In fact, we generalize the range of the expected degree d = np based on their proof from d = c to d c and d k−1 = o(n) for some large enough constant c with better analysis.…”

Section: Preliminariesmentioning

confidence: 53%

“…We remark that d k−1 = o(n) is a necessary condition in the proof of Theorem 1.1 and Theorem 1.3. The way to obtain the upper bound of the largest k-matching in Theorem 1.1 is similar with the one when d is a large constant in [5]. For k = 2, it coincides with the upper bound um [2,3].…”

Section: Introductionmentioning

confidence: 55%

“…when the expected degree d = np = Ω(1) and d = o(n), further um 2 (G) log h n with h = 1 1−p when p > n −ǫ for any given ǫ > 0. In a specially sparse case when d is a sufficiently large constant, for any fixed integer k 2 and G ∈ G(n, p), Kang and Manggala [5] improved the upper bound of um k (G) to um k (G) (1 + o(1)) kn log d 2d k−1 . It is speculated in [5] that the upper bound here is close to the correct value of um k (G) when k 2.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, for any fixed integer k 2 and G ∈ G(n, p), as a further step of [2,3,5,7], we consider the bounds of um k (G). In fact, we mainly investigate e(M k ) of any maximal k-matching M k in G ∈ G(n, p).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Large induced distance matchings in certain sparse random graphs

Tian¹,

Sun²,

Liu³

2022

Preprint

View full text Add to dashboard Cite

For a fixed integer k 2, let G ∈ G(n, p) be a simple connected graph on n → ∞ vertices with the expected degree d = np satisfying d c and d k−1 = o(n) for some large enough constant c. We show that the asymptotical size of any maximal collection of edges M in G such that no two edges in M are within distance k, which is called a distance k-matching, is between (k−1)n log d 4d k−1 and kn log d 2d k−1 . We also design a randomized greedy algorithm to generate one large distance k-matching in G with asymptotical size kn log d 4d k−1 . Our results partially generalize the results on the size of the largest distance k-matchings from the case k = 2 or d = c for some large constant c.

show abstract

Section: Preliminariesmentioning

confidence: 53%

Section: Introductionmentioning

confidence: 55%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Large induced distance matchings in certain sparse random graphs

Tian¹,

Sun²,

Liu³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…In particular, there has been considerable interest in χ(L(G) t ) (where χ(H) denotes the chromatic number of H), especially for G of bounded maximum degree. For t = 1, this is the usual chromatic index of G; for t = 2, it is known as the strong chromatic index of G, and is associated with a more famous problem of Erdős and Nešetřil [9]; for t > 2, the parameter is referred to as the distance-t chromatic index, with the study of bounded degree graphs initiated in [13]. We note that the output of Theorem 8 may be directly used as input to a recent result [11] related to Reed's conjecture [15] to bound χ(L(G) t ).…”

Section: Introductionmentioning

confidence: 99%

Maximizing Line Subgraphs of Diameter at Most t

Cambie¹,

Batenburg²,

Verclos³

et al. 2022

SIAM J. Discrete Math.

Self Cite

View full text Add to dashboard Cite

We wish to bring attention to a natural but slightly hidden problem, posed by Erdős and Nešetřil in the late 1980s, an edge version of the degree-diameter problem. Our main result is that, for any graph of maximum degree ∆ with more than 1.5∆ t edges, its line graph must have diameter larger than t. In the case where the graph contains no cycle of length 2t + 1, we can improve the bound on the number of edges to one that is exact for t ∈ {1, 2, 3, 4, 6}. In the case ∆ = 3 and t = 3, we obtain an exact bound. Our results also have implications for the related problem of bounding the distance-t chromatic index, t > 2; in particular, for this we obtain an upper bound of 1.941∆ t for graphs of large enough maximum degree ∆, markedly improving upon earlier bounds for this parameter.

show abstract