Parity vertex colouring of graphs

Borowiecki, Piotr; Budajová, Kristína; Jendroľ, Stanislav; Krajci, Stanislav

doi:10.7151/dmgt.1537

Cited by 9 publications

(15 citation statements)

References 11 publications

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“…Odd colorings with respect to paths of graphs have been recently studied in [4,10], independently from our work. In these papers, they are called parity vertex colorings.…”

Section: Introductionmentioning

confidence: 96%

Unique-Maximum and Conflict-Free Coloring for Hypergraphs and Tree Graphs

Cheilaris¹,

Keszegh²,

Pálvölgyi³

2013

SIAM J. Discrete Math.

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Abstract. We investigate the relationship between two kinds of vertex colorings of hypergraphs: unique-maximum colorings and conflict-free colorings. In a unique-maximum coloring, the colors are ordered, and in every hyperedge of the hypergraph the maximum color in the hyperedge occurs in only one vertex of the hyperedge. In a conflict-free coloring, in every hyperedge of the hypergraph there exists a color in the hyperedge that occurs in only one vertex of the hyperedge. We consider the corresponding unique-maximum and conflict-free chromatic numbers and investigate their relationship in arbitrary hypergraphs. Then, we concentrate on hypergraphs that are induced by simple paths in tree graphs. Introduction.A hypergraph H is a pair (V, E), where E (the hyperedge set) is a family of non-empty subsets of V (the vertex set). A vertex coloring of a hypergraph H = (V, E) is a function C : V → Z + . A hypergraph is a generalization of a graph. Therefore, it is natural to consider how to generalize proper vertex coloring of a graph to a vertex coloring of a hypergraph. (In a proper vertex coloring of a graph, any two vertices neighboring with an edge in the graph have to be assigned different colors by the coloring function C.) Vertex coloring in hypergraphs can be defined in many ways, so that when restricting the definition to simple graphs, it coincides with proper graph coloring.At one extreme, it is only required that the vertices of each hyperedge are not all colored with the same color (except for singleton hyperedges). This is called a nonmonochromatic coloring of a hypergraph. The minimum number of colors necessary to color in such a way a hypergraph H is the (non-monochromatic) chromatic number of H, denoted by χ(H).At the other extreme, we can require that the vertices of each hyperedge are all colored with different colors. This is called a colorful or rainbow coloring of H and we have the corresponding rainbow chromatic number of H, denoted by χ rb (H).In this paper we investigate the following two types of vertex colorings of hypergraphs that are between the above two extremes. Definition 1.1. A unique-maximum coloring of H = (V, E) with k colors is a function C : V → {1, . . . , k} such that for each e ∈ E the maximum color in e occurs exactly once on the vertices of e. The minimum k for which a hypergraph H has a unique-maximum coloring with k colors is called the unique-maximum chromatic number of H and is denoted by χ um (H). Definition 1.2. A conflict-free coloring of H = (V, E) with k colors is a function C : V → {1, . . . , k} such that for each e ∈ E there is a color in e that occurs exactly once on the vertices of e. The minimum k for which a hypergraph H has a conflict-free coloring with k colors is called the conflict-free chromatic number of H and is denoted by χ cf (H).We also introduce a new coloring, that proves useful in showing lower bounds, and that could be of independent interest.

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“…Odd colorings with respect to paths of graphs have been recently studied in [4,10], independently from our work. In these papers, they are called parity vertex colorings.…”

Section: Introductionmentioning

confidence: 96%

Unique-Maximum and Conflict-Free Coloring for Hypergraphs and Tree Graphs

Cheilaris¹,

Keszegh²,

Pálvölgyi³

2013

SIAM J. Discrete Math.

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“…A parity vertex coloring of a graph G is a vertex coloring such that each path in G contains some color odd number of times. For a study of parity vertex and (similarly defined) edge colorings, the reader is referred to [1,2]. A vertex ranking of G is a proper vertex coloring by a linearly ordered set of colors such that every path between vertices of the same color contains some vertex of a higher color.…”

Section: Introductionmentioning

confidence: 99%

Parity vertex colorings of binomial trees

Gregor¹,

Škrekovski²

2012

Discuss. Math. Graph Theory

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“…The (rooted) binomial tree B k with 2 k−1 vertices is defined as follows: B 1 is a single vertex; for k > 1, B k consists of two disjoint copies of B k−1 and an edge between their two roots, where the root of B k is the root of the first copy. These trees are used in [1,14]. The binomial tree B k is another tree class for which the choice of the balanced edge of each generated subtree is unique in Algorithm 1.…”

Section: An Algorithm For the Upper Boundmentioning

confidence: 99%

Conflict-free connection of trees

Chang

et al. 2018

J Comb Optim

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We study the conflict-free connection coloring of trees, which is also the conflict-free coloring of the so-called edge-path hypergraphs of trees. We first prove that for a tree T of order n, cf c(T ) ≥ cf c(P n ) = ⌈log 2 n⌉, which completely confirms the conjecture of Li and Wu. We then present a sharp upper bound for the conflict-free connection number of trees by a simple algorithm. Furthermore, we show that the conflict-free connection number of the binomial tree with 2 k−1 vertices is k − 1. At last, we study trees which are cf c-critical, and prove that if a tree T is cf c-critical, then the conflict-free connection coloring of T is equivalent to the edge ranking of T .

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Parity vertex colouring of graphs

Cited by 9 publications

References 11 publications

Unique-Maximum and Conflict-Free Coloring for Hypergraphs and Tree Graphs

Unique-Maximum and Conflict-Free Coloring for Hypergraphs and Tree Graphs

Parity vertex colorings of binomial trees

Conflict-free connection of trees

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