4-chromatic graphs with large odd girth

Ngọc, Nguyễn Văn; Tuza, Źsolt

doi:10.1016/0012-365x(94)00221-4

Cited by 18 publications

(10 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Definition 3. [8,9,11,12] Given a graph G and an integer m ≥ 1, the m-Mycielskian of G, denoted µ m (G), is defined to be the graph with vertex set (V (G) × {0, 1, . .…”

Section: Definitionsmentioning

confidence: 99%

See 1 more Smart Citation

A Note on the Immersion Number of Generalized Mycielski Graphs

Collins¹,

Heenehan²,

McDonald³

2021

Preprint

View full text Add to dashboard Cite

The immersion number of a graph G, denoted im(G), is the largest t such that G has a Kt-immersion. In this note we are interested in determining the immersion number of the m-Mycielskian of G, denoted µm(G). Given the immersion number of G we provide a lower bound for im(µm(G)). To do this we introduce the "distinct neighbor property" of immersions. We also include examples of classes of graphs where im(µm(G)) exceeds the lower bound. We conclude with a conjecture about im(µm(Kt)).

show abstract

“…Definition 3. [8,9,11,12] Given a graph G and an integer m ≥ 1, the m-Mycielskian of G, denoted µ m (G), is defined to be the graph with vertex set (V (G) × {0, 1, . .…”

Section: Definitionsmentioning

confidence: 99%

“…Hence Mycielski graphs provide examples of triangle-free graphs with arbitrarily high chromatic number. The Mycielski construction was generalized by Stiebitz [9] (see also [8]) and independently by Van Ngoc [11] (see also [12]). The general construction was also described as the "cone over G" by Tardif [10].…”

Section: Introductionmentioning

confidence: 99%

A Note on the Immersion Number of Generalized Mycielski Graphs

Collins¹,

Heenehan²,

McDonald³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Starting with a graph G of any odd girth, Van Ngoc and Tuza [14] and independently Youngs [16] have used a construction similar to the Mycielski construction to create 4-chromatic graphs of arbitrarily large odd girth 2k + 1 where k ≥ 2. We denote this generalized construction by M k (G).…”

Section: Cores With Fixed Vertex Factorsmentioning

confidence: 99%

“…where G i is a connected core with odd girth at least 2k i + 1 and M k i (G i ) is the generalized Mycielski construction on G i to get a graph of odd girth 2k i + 1 for k i ≥ 2, see [14,16].…”

Section: Introductionmentioning

confidence: 99%

Retracts of box products with odd‐angulated factors

Che

Collins

2006

Journal of Graph Theory

View full text Add to dashboard Cite

Let G be a connected graph with odd girth 2κ + 1. Then G is a (2κ + 1)-angulated graph if every two vertices of G are connected by a path such that each edge of the path is in some (2κ + 1)-cycle. We prove that if G is (2κ + 1)-angulated, and H is connected with odd girth at least 2κ + 3, then any retract of the box (or Cartesian) product G H is S T where S is a retract of G and T is a connected subgraph of H. A graph G is strongly (2κ + 1)-angulated if any two vertices of G are connected by a sequence of (2κ + 1)-cycles with consecutive cycles sharing at least one edge. We prove that if G is strongly (2κ + 1)-angulated, and H is connected with odd girth at least 2κ + 1, then any retract of G H is S T where S is a retract of G and T is a connected subgraph of H or |V(S)| = 1 and T is a retract of H. These two results improve theorems on weakly and strongly triangulated graphs by Nowakowski and Rival [Disc Math 70 (1988), 169-184]. As a corollary, we get that the core of the box product of two strongly (2κ + 1)-angulated cores must be either one of the factors or the box product itself. Furthermore, if G is a strongly (2κ + 1)-angulated core, then either G n is a core for all positive integers n, or the core of G n is G for all positive integers n. In the latter case, G is homomorphically equivalent to a Journal of Graph Theory © 2006 Wiley Periodicals, Inc. 24 RETRACTS OF ODD-ANGULATED GRAPHS 25normal Cayley graph [Larose, Laviolette, Tardiff, European J Combin 19 (1998), 867-881]. In particular, let G be a strongly (2κ + 1)-angulated core such that either G is not vertex-transitive, or G is vertex-transitive and any two maximum independent sets have non-empty intersection. Then G n is a core for any positive integer n. On the other hand, let G i be a (2κ i + 1)-angulated core for 1 ≤ i ≤ n where κ 1 < κ 2 < · · · < κ n . If G i has a vertex that is fixed under any automorphism for 1 ≤ i ≤ n − 1, or G i is vertex-transitive such that any two maximum independent sets have non-empty intersection for 1 ≤ i ≤ n − 1, then n i=1 G i is a core. We then apply the results to construct cores that are box products with Mycielski construction factors or with odd graph factors. We also show that K(r, 2r + 1) C 2l+1 is a core for any integers l ≥ r ≥ 2. It is open whether K(r, 2r + 1) C 2l+1 is a core for r > l ≥ 2.

show abstract

“…To this day, no combinatorial proof of Theorem 1.1 is known (see [8, pp. 133]), except for the case k = 4 [17]. At the end of their paper, Van Ngoc and Tuza [17] propose the following problem:…”

Section: Introductionmentioning

confidence: 99%

Generalised Mycielski Graphs and the Borsuk–Ulam Theorem

Müller¹,

Stehlı́k²

2019

Electron. J. Combin.

View full text Add to dashboard Cite

Stiebitz determined the chromatic number of generalised Mycielski graphs using the topological method of Lovász, which invokes the Borsuk-Ulam theorem. Van Ngoc and Tuza used elementary combinatorial arguments to prove Stiebitz's theorem for 4-chromatic generalised Mycielski graphs, and asked if there is also an elementary combinatorial proof for higher chromatic number. We answer their question by showing that Stiebitz's theorem can be deduced from a version of Fan's combinatorial lemma. Our proof uses topological terminology, but is otherwise completely discrete and could be rewritten to avoid topology altogether. However, doing so would be somewhat artificial, because we also show that Stiebitz's theorem is equivalent to the Borsuk-Ulam theorem.

show abstract

4-chromatic graphs with large odd girth

Cited by 18 publications

References 6 publications

A Note on the Immersion Number of Generalized Mycielski Graphs

A Note on the Immersion Number of Generalized Mycielski Graphs

Retracts of box products with odd‐angulated factors

Generalised Mycielski Graphs and the Borsuk–Ulam Theorem

Contact Info

Product

Resources

About