On the edge-density of 4-critical graphs

Farzad, Babak; Molloy, Michael

doi:10.1007/s00493-009-2267-y

Cited by 6 publications

(5 citation statements)

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“…Even this seemingly simple lower bound is still unresolved; the best known result is a recent proof by Farzad and Molloy [7] that this inequality is true provided that the subgraph induced by vertices of degree 3 is connected.…”

Section: Edge Density Of Undirected Graphsmentioning

confidence: 95%

The edge density of critical digraphs

Hoshino

Kawarabayashi

2014

Combinatorica

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Let χ(G) denote the chromatic number of a graph G. We say that G is k-critical if χ(G) = k and χ(H) < k for every proper subgraph H ⊂ G. Over the years, many properties of kcritical graphs have been discovered, including improved upper and lower bounds for ||G||, the number of edges in a k-critical graph, as a function of |G|, the number of vertices. In this note, we analyze this edge density problem for directed graphs, where the chromatic number χ(D) of a digraph D is defined to be the fewest number of colours needed to colour the vertices of D so that each colour class induces an acyclic subgraph. For each k ≥ 3, we construct an infinite family of sparse k-critical digraphs for which ||D|| < ( k 2 −k+1 2 )|D| and an infinite family of dense k-critical digraphs for which ||D|| > 1 2 − 1 2 k−1 |D| 2 . One corollary of our results is an explicit construction of an infinite family of k-critical digraphs of digirth l, for any pair of integers k, l ≥ 3. This extends a result by Bokal et al. who used a probabilistic approach to demonstrate the existence of one such digraph.

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Section: Edge Density Of Undirected Graphsmentioning

confidence: 95%

The edge density of critical digraphs

Hoshino

Kawarabayashi

2014

Combinatorica

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“…It is one half of Problem P1 in [30, P. 347]. Recently, Farzad and Molloy [10] have found the minimum number of edges in 4-critical n-vertex graphs in which the set of vertices of degree 3 induces a connected subgraph.…”

Section: Introductionmentioning

confidence: 99%

Ore's conjecture on color-critical graphs is almost true

Kostochka

Yancey

2014

Journal of Combinatorial Theory, Series B

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“…Note that Conjecture 2 is equivalent to the case n ≡ 1 ( mod k − 1) of Conjecture 3. Some lower bounds on f k (n) were obtained in [12,28,16,24,25,15]. Recently, the authors [26] proved Conjecture 2 valid.…”

Section: Introductionmentioning

confidence: 99%