Upper oriented chromatic number of undirected graphs and oriented colorings of product graphs

Éric Sopena

Displaying similar documents to “Upper oriented chromatic number of undirected graphs and oriented colorings of product graphs”

On 𝓕-independence in graphs

Frank Göring, Jochen Harant, Dieter Rautenbach, Ingo Schiermeyer (2009)

Discussiones Mathematicae Graph Theory

Similarity:

Let be a set of graphs and for a graph G let $α_{} (G)$ and $α *_{} (G)$ denote the maximum order of an induced subgraph of G which does not contain a graph in as a subgraph and which does not contain a graph in as an induced subgraph, respectively. Lower bounds on $α_{} (G)$ and $α *_{} (G)$ are presented.

On subgraphs without large components

Glenn G. Chappell, John Gimbel (2017)

Mathematica Bohemica

Similarity:

We consider, for a positive integer $k$ , induced subgraphs in which each component has order at most $k$ . Such a subgraph is said to be $k$ -divided. We show that finding large induced subgraphs with this property is NP-complete. We also consider a related graph-coloring problem: how many colors are required in a vertex coloring in which each color class induces a $k$ -divided subgraph. We show that the problem of determining whether some given number of colors suffice is NP-complete, even for...

On characterization of uniquely 3-list colorable complete multipartite graphs

Yancai Zhao, Erfang Shan (2010)

Discussiones Mathematicae Graph Theory

Similarity:

For each vertex v of a graph G, if there exists a list of k colors, L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k-list colorable graph. Ghebleh and Mahmoodian characterized uniquely 3-list colorable complete multipartite graphs except for nine graphs: $K_{2, 2, r}$ r ∈ 4,5,6,7,8, $K_{2, 3, 4}$ , $K_{1 * 4, 4}$ , $K_{1 * 4, 5}$ , $K_{1 * 5, 4}$ . Also, they conjectured that the nine graphs are not U3LC graphs. After that, except for $K_{2, 2, r}$ r ∈ 4,5,6,7,8, the others have been proved not...

On distinguishing and distinguishing chromatic numbers of hypercubes

Werner Klöckl (2008)

Discussiones Mathematicae Graph Theory

Similarity:

The distinguishing number D(G) of a graph G is the least integer d such that G has a labeling with d colors that is not preserved by any nontrivial automorphism. The restriction to proper labelings leads to the definition of the distinguishing chromatic number $χ_{D} (G)$ of G. Extending these concepts to infinite graphs we prove that $D (Q_{ℵ} ₀) = 2$ and $χ_{D} (Q_{ℵ} ₀) = 3$ , where $Q_{ℵ} ₀$ denotes the hypercube of countable dimension. We also show that $χ_{D} (Q ₄) = 4$ , thereby completing the investigation of finite hypercubes with respect to $χ_{D}$ . Our...

Remarks on $D$ -integral complete multipartite graphs

Pavel Híc, Milan Pokorný (2016)

Czechoslovak Mathematical Journal

Similarity:

A graph is called distance integral (or $D$ -integral) if all eigenvalues of its distance matrix are integers. In their study of $D$ -integral complete multipartite graphs, Yang and Wang (2015) posed two questions on the existence of such graphs. We resolve these questions and present some further results on $D$ -integral complete multipartite graphs. We give the first known distance integral complete multipartite graphs $K_{p_{1}, p_{2}, p_{3}}$ with $p_{1} < p_{2} < p_{3}$ , and $K_{p_{1}, p_{2}, p_{3}, p_{4}}$ with $p_{1} < p_{2} < p_{3} < p_{4}$ , as well as the infinite classes of distance integral...

Domination and independence subdivision numbers of graphs

Teresa W. Haynes, Sandra M. Hedetniemi, Stephen T. Hedetniemi (2000)

Discussiones Mathematicae Graph Theory

Similarity:

The domination subdivision number $s d_{γ} (G)$ of a graph is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number. Arumugam showed that this number is at most three for any tree, and conjectured that the upper bound of three holds for any graph. Although we do not prove this interesting conjecture, we give an upper bound for the domination subdivision number for any graph G in terms of the minimum degrees of...

Neighbor sum distinguishing list total coloring of IC-planar graphs without 5-cycles

Donghan Zhang (2022)

Czechoslovak Mathematical Journal

Similarity:

Let $G = (V (G), E (G))$ be a simple graph and $E_{G} (v)$ denote the set of edges incident with a vertex $v$ . A neighbor sum distinguishing (NSD) total coloring $φ$ of $G$ is a proper total coloring of $G$ such that $\sum_{z \in E_{G} (u) \cup {u}} φ (z) \neq \sum_{z \in E_{G} (v) \cup {v}} φ (z)$ for each edge $u v \in E (G)$ . Pilśniak and Woźniak asserted in 2015 that each graph with maximum degree $Δ$ admits an NSD total $(Δ + 3)$ -coloring. We prove that the list version of this conjecture holds for any IC-planar graph with $Δ \geq 11$ but without $5$ -cycles by applying the Combinatorial Nullstellensatz.

Intrinsic linking and knotting are arbitrarily complex

Erica Flapan, Blake Mellor, Ramin Naimi (2008)

Fundamenta Mathematicae

Similarity:

We show that, given any n and α, any embedding of any sufficiently large complete graph in ℝ³ contains an oriented link with components Q₁, ..., Qₙ such that for every i ≠ j, $| l k (Q_{i}, Q_{j}) | \geq α$ and $| a ₂ (Q_{i}) | \geq α$ , where $a ₂ (Q_{i})$ denotes the second coefficient of the Conway polynomial of $Q_{i}$ .

On the total k-domination number of graphs

Adel P. Kazemi (2012)

Discussiones Mathematicae Graph Theory

Similarity:

Let k be a positive integer and let G = (V,E) be a simple graph. The k-tuple domination number $γ_{\times k} (G)$ of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V, $| N_{G} [v] \cap S | \geq k$ . Also the total k-domination number $γ_{\times k, t} (G)$ of G is the minimum cardinality of a total k -dominating set S, a set that for every vertex v ∈ V, $| N_{G} (v) \cap S | \geq k$ . The k-transversal number τₖ(H) of a hypergraph H is the minimum size of a subset S ⊆ V(H) such that |S ∩e | ≥ k for every edge e ∈ E(H). We know that for...

Upper bounds for the domination numbers of toroidal queens graphs

Christina M. Mynhardt (2003)

Discussiones Mathematicae Graph Theory

Similarity:

We determine upper bounds for $γ (Q n^{t})$ and $i (Q ₙ^{t})$ , the domination and independent domination numbers, respectively, of the graph $Q ₙ^{t}$ obtained from the moves of queens on the n×n chessboard drawn on the torus.

A note on the independent domination number versus the domination number in bipartite graphs

Shaohui Wang, Bing Wei (2017)

Czechoslovak Mathematical Journal

Similarity:

Let $γ (G)$ and $i (G)$ be the domination number and the independent domination number of $G$ , respectively. Rad and Volkmann posted a conjecture that $i (G) / γ (G) \leq Δ (G) / 2$ for any graph $G$ , where $Δ (G)$ is its maximum degree (see N. J. Rad, L. Volkmann (2013)). In this work, we verify the conjecture for bipartite graphs. Several graph classes attaining the extremal bound and graphs containing odd cycles with the ratio larger than $Δ (G) / 2$ are provided as well.

On the diameter of the intersection graph of a finite simple group

Xuanlong Ma (2016)

Czechoslovak Mathematical Journal

Similarity:

Let $G$ be a finite group. The intersection graph $Δ_{G}$ of $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of $G$ , and two distinct vertices $X$ and $Y$ are adjacent if $X \cap Y \neq 1$ , where $1$ denotes the trivial subgroup of order $1$ . A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters...

Edge-colouring of graphs and hereditary graph properties

Samantha Dorfling, Tomáš Vetrík (2016)

Czechoslovak Mathematical Journal

Similarity:

Edge-colourings of graphs have been studied for decades. We study edge-colourings with respect to hereditary graph properties. For a graph $G$ , a hereditary graph property $𝒫$ and $l \geq 1$ we define $χ_{𝒫, l}^{'} (G)$ to be the minimum number of colours needed to properly colour the edges of $G$ , such that any subgraph of $G$ induced by edges coloured by (at most) $l$ colours is in $𝒫$ . We present a necessary and sufficient condition for the existence of $χ_{𝒫, l}^{'} (G)$ . We focus on edge-colourings of graphs with respect to the hereditary...

Edit distance measure for graphs

Tomasz Dzido, Krzysztof Krzywdziński (2015)

Czechoslovak Mathematical Journal

Similarity:

In this paper, we investigate a measure of similarity of graphs similar to the Ramsey number. We present values and bounds for $g (n, l)$ , the biggest number $k$ guaranteeing that there exist $l$ graphs on $n$ vertices, each two having edit distance at least $k$ . By edit distance of two graphs $G$ , $F$ we mean the number of edges needed to be added to or deleted from graph $G$ to obtain graph $F$ . This new extremal number $g (n, l)$ is closely linked to the edit distance of graphs. Using probabilistic methods we show...

Paired domination in prisms of graphs

Christina M. Mynhardt, Mark Schurch (2011)

Discussiones Mathematicae Graph Theory

Similarity:

The paired domination number $γ_{p r} (G)$ of a graph G is the smallest cardinality of a dominating set S of G such that ⟨S⟩ has a perfect matching. The generalized prisms πG of G are the graphs obtained by joining the vertices of two disjoint copies of G by |V(G)| independent edges. We provide characterizations of the following three classes of graphs: $γ_{p r} (π G) = 2 γ_{p r} (G)$ for all πG; $γ_{p r} (K ₂ ☐ G) = 2 γ_{p r} (G)$ ; $γ_{p r} (K ₂ ☐ G) = γ_{p r} (G)$ .

Nonempty intersection of longest paths in a graph with a small matching number

Fuyuan Chen (2015)

Czechoslovak Mathematical Journal

Similarity:

A maximum matching of a graph $G$ is a matching of $G$ with the largest number of edges. The matching number of a graph $G$ , denoted by $α^{'} (G)$ , is the number of edges in a maximum matching of $G$ . In 1966, Gallai conjectured that all the longest paths of a connected graph have a common vertex. Although this conjecture has been disproved, finding some nice classes of graphs that support this conjecture is still very meaningful and interesting. In this short note, we prove that Gallai’s conjecture...

Classes of hypergraphs with sum number one

Hanns-Martin Teichert (2000)

Discussiones Mathematicae Graph Theory

Similarity:

A hypergraph ℋ is a sum hypergraph iff there are a finite S ⊆ ℕ⁺ and d̲,d̅ ∈ ℕ⁺ with 1 < d̲ < d̅ such that ℋ is isomorphic to the hypergraph $ℋ_{d ̲, d ̅} (S) = (V,)$ where V = S and $= e \subseteq S : d ̲ < | e | < d ̅ \land \sum_{v \in e} v \in S$ . For an arbitrary hypergraph ℋ the sum number(ℋ ) is defined to be the minimum number of isolatedvertices $w ₁, . . ., w_{σ} \notin V$ such that $ℋ \cup w ₁, . . ., w_{σ}$ is a sum hypergraph. For graphs it is known that cycles Cₙ and wheels Wₙ have sum numbersgreater than one. Generalizing these graphs we prove for the hypergraphs ₙ and ₙ that under a certain condition...