An Upper Bound on the Fractional Chromatic Number of Triangle-Free Subcubic Graphs

Liu, Chun-Hung

doi:10.1137/120900678

“…By combining Corollary 5.5, Lemma 5.6 (for l = 2), Lemma 4.4 and Liu's bound [31] similarly as in Section 5.2, we get the following corollary.…”

Section: K-path In Undirected Subcubic Graphsmentioning

confidence: 61%

“…By Lemma 4.4, such an algorithm would imply an algorithm for (k, l)-tree running in time O * (2 l ) = O * (1.562 k 1.281 l ). However, we may use an earlier result of Liu [31], which gives a bound of 43 15 . Liu's proof is inductive and can be turned to a polynomial-time algorithm which finds a (516 : 180)-coloring [32].…”

Section: Subcubic Graphs Of Large Girthmentioning

confidence: 99%

“…For example, since χ f ≤ 14/5 for triangle-free subcubic graphs [13], by plugging it into Corollary 5.8 and using the triangle-removing trick from Section 5.3, one gets an O * (1.576 k )-time algorithm for k-Path in subcubic graphs (which is slightly worse than the bound of Corollary 5.7, but avoids the use of the complicated Liu's algorithm [31]). …”

Section: Solving (K L)-tree In Graphs With Small Vector Chromatic Numentioning

confidence: 99%

See 1 more Smart Citation

Spotting Trees with Few Leaves

Björklund

¹

,

Kamat

²

,

Kowalik

³

et al. 2017

SIAM J. Discrete Math.

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We show two results related to the Hamiltonicity and k-Path algorithms in undirected graphs by Björklund [FOCS'10], and Björklund et al., [arXiv'10]. First, we demonstrate that the technique used can be generalized to finding some k-vertex tree with l leaves in an n-vertex undirected graph in O * (1.657 k 2 l/2 ) time. It can be applied as a subroutine to solve the k-Internal Spanning Tree (k-IST) problem in O * (min(3.455 k , 1.946 n )) time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time we break the natural barrier of O * (2 n ). Second, we show that the iterated random bipartition employed by the algorithm can be improved whenever the host graph admits a vertex coloring with few colors; it can be an ordinary proper vertex coloring, a fractional vertex coloring, or a vector coloring. In effect, we show improved bounds for k-Path and Hamiltonicity in any graph of maximum degree ∆ = 4, . . . , 12 or with vector chromatic number at most 8.

show abstract

“…By Lemma 4.4, such an algorithm would imply an algorithm for (k, l)-tree running in time O * (2 9 14 k+ 5 14 l ) = O * (1.562 k 1.281 l ). However, we may use an earlier result of Liu [31], which gives a bound of 43 15 . Liu's proof is inductive and can be turned to a polynomial-time algorithm which finds a (516 : 180)-coloring [32].…”

Section: Subcubic Graphs Of Large Girthmentioning

confidence: 99%

“…Let us observe that thanks to Theorem 4.1, we can apply Corollary 5.8 whenever we know that a chromatic number or the fractional chromatic number is low, even if no efficient algorithm for finding the coloring is known. For example, since χ f ≤ 14/5 for triangle-free subcubic graphs [13], by plugging it into Corollary 5.8 and using the triangle-removing trick from Section 5.3, one gets an O * (1.576 k )-time algorithm for k-Path in subcubic graphs (which is slightly worse than the bound of Corollary 5.7, but avoids the use of the complicated Liu's algorithm [31]).…”

Section: Solving (K L)-tree In Graphs With Small Vector Chromatic Numentioning

confidence: 99%

Spotting Trees with Few Leaves

Björklund

¹

,

Kamat

²

,

Kowalik

³

et al. 2015

Automata, Languages, and Programming

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We show two results related to the Hamiltonicity and k-Path algorithms in undirected graphs by Björklund [FOCS'10], and Björklund et al., [arXiv'10]. First, we demonstrate that the technique used can be generalized to finding some k-vertex tree with l leaves in an n-vertex undirected graph in O * (1.657 k 2 l/2 ) time. It can be applied as a subroutine to solve the k-Internal Spanning Tree (k-IST) problem in O * (min(3.455 k , 1.946 n )) time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time we break the natural barrier of O * (2 n ). Second, we show that the iterated random bipartition employed by the algorithm can be improved whenever the host graph admits a vertex coloring with few colors; it can be an ordinary proper vertex coloring, a fractional vertex coloring, or a vector coloring. In effect, we show improved bounds for k-Path and Hamiltonicity in any graph of maximum degree ∆ = 4, . . . , 12 or with vector chromatic number at most 8.

show abstract