Radio k-labelings for Cartesian products of graphs

Mustapha Kchikech; Riadh Khennoufa; Olivier Togni

Displaying similar documents to “Radio k-labelings for Cartesian products of graphs”

Upper bounds on the b-chromatic number and results for restricted graph classes

Mais Alkhateeb, Anja Kohl (2011)

Discussiones Mathematicae Graph Theory

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A b-coloring of a graph G by k colors is a proper vertex coloring such that every color class contains a color-dominating vertex, that is, a vertex having neighbors in all other k-1 color classes. The b-chromatic number $χ_{b} (G)$ is the maximum integer k for which G has a b-coloring by k colors. Moreover, the graph G is called b-continuous if G admits a b-coloring by k colors for all k satisfying $χ (G) \leq k \leq χ_{b} (G)$ . In this paper, we establish four general upper bounds on $χ_{b} (G)$ . We present results on the b-chromatic...

Acyclic reducible bounds for outerplanar graphs

Mieczysław Borowiecki, Anna Fiedorowicz, Mariusz Hałuszczak (2009)

Discussiones Mathematicae Graph Theory

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For a given graph G and a sequence ₁, ₂,..., ₙ of additive hereditary classes of graphs we define an acyclic (₁, ₂,...,Pₙ)-colouring of G as a partition (V₁, V₂,...,Vₙ) of the set V(G) of vertices which satisfies the following two conditions: 1. $G [V_{i}] \in_{i}$ for i = 1,...,n, 2. for every pair i,j of distinct colours the subgraph induced in G by the set of edges uv such that $u \in V_{i}$ and $v \in V_{j}$ is acyclic. A class R = ₁ ⊙ ₂ ⊙ ... ⊙ ₙ is defined as the set of the graphs having an acyclic (₁, ₂,...,Pₙ)-colouring....

Maximal k-independent sets in graphs

Mostafa Blidia, Mustapha Chellali, Odile Favaron, Nacéra Meddah (2008)

Discussiones Mathematicae Graph Theory

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A subset of vertices of a graph G is k-independent if it induces in G a subgraph of maximum degree less than k. The minimum and maximum cardinalities of a maximal k-independent set are respectively denoted iₖ(G) and βₖ(G). We give some relations between βₖ(G) and $β_{j} (G)$ and between iₖ(G) and $i_{j} (G)$ for j ≠ k. We study two families of extremal graphs for the inequality i₂(G) ≤ i(G) + β(G). Finally we give an upper bound on i₂(G) and a lower bound when G is a cactus.

Fall coloring of graphs I

Rangaswami Balakrishnan, T. Kavaskar (2010)

Discussiones Mathematicae Graph Theory

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A fall coloring of a graph G is a proper coloring of the vertex set of G such that every vertex of G is a color dominating vertex in G (that is, it has at least one neighbor in each of the other color classes). The fall coloring number $χ_{f} (G)$ of G is the minimum size of a fall color partition of G (when it exists). Trivially, for any graph G, $χ (G) \leq χ_{f} (G)$ . In this paper, we show the existence of an infinite family of graphs G with prescribed values for χ(G) and $χ_{f} (G)$ . We also obtain the smallest non-fall...

3-consecutive c-colorings of graphs

Csilla Bujtás, E. Sampathkumar, Zsolt Tuza, M.S. Subramanya, Charles Dominic (2010)

Discussiones Mathematicae Graph Theory

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A 3-consecutive C-coloring of a graph G = (V,E) is a mapping φ:V → ℕ such that every path on three vertices has at most two colors. We prove general estimates on the maximum number ${(χ ̅)}_{3 C C} (G)$ of colors in a 3-consecutive C-coloring of G, and characterize the structure of connected graphs with ${(χ ̅)}_{3 C C} (G) \geq k$ for k = 3 and k = 4.

Total domination versus paired domination

Oliver Schaudt (2012)

Discussiones Mathematicae Graph Theory

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A dominating set of a graph G is a vertex subset that any vertex of G either belongs to or is adjacent to. A total dominating set is a dominating set whose induced subgraph does not contain isolated vertices. The minimal size of a total dominating set, the total domination number, is denoted by γₜ. The maximal size of an inclusionwise minimal total dominating set, the upper total domination number, is denoted by Γₜ. A paired dominating set is a dominating set whose induced subgraph has...

On generalized shift graphs

Christian Avart, Tomasz Łuczak, Vojtěch Rödl (2014)

Fundamenta Mathematicae

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In 1968 Erdős and Hajnal introduced shift graphs as graphs whose vertices are the k-element subsets of [n] = 1,...,n (or of an infinite cardinal κ ) and with two k-sets $A = a ₁, . . ., a_{k}$ and $B = b ₁, . . ., b_{k}$ joined if $a ₁ < a ₂ = b ₁ < a ₃ = b ₂ < \dots < a_{k} = b_{k - 1} < b_{k}$ . They determined the chromatic number of these graphs. In this paper we extend this definition and study the chromatic number of graphs defined similarly for other types of mutual position with respect to the underlying ordering. As a consequence of our result, we show the existence of a graph with...

Domination in partitioned graphs

Zsolt Tuza, Preben Dahl Vestergaard (2002)

Discussiones Mathematicae Graph Theory

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Let V₁, V₂ be a partition of the vertex set in a graph G, and let $γ_{i}$ denote the least number of vertices needed in G to dominate $V_{i}$ . We prove that γ₁+γ₂ ≤ [4/5]|V(G)| for any graph without isolated vertices or edges, and that equality occurs precisely if G consists of disjoint 5-paths and edges between their centers. We also give upper and lower bounds on γ₁+γ₂ for graphs with minimum valency δ, and conjecture that γ₁+γ₂ ≤ [4/(δ+3)]|V(G)| for δ ≤ 5. As δ gets large, however, the largest...

Radio numbers for generalized prism graphs

Paul Martinez, Juan Ortiz, Maggy Tomova, Cindy Wyels (2011)

Discussiones Mathematicae Graph Theory

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A radio labeling is an assignment c:V(G) → N such that every distinct pair of vertices u,v satisfies the inequality d(u,v) + |c(u)-c(v)| ≥ diam(G) + 1. The span of a radio labeling is the maximum value. The radio number of G, rn(G), is the minimum span over all radio labelings of G. Generalized prism graphs, denoted $Z_{n, s}$ , s ≥ 1, n ≥ s, have vertex set (i,j) | i = 1,2 and j = 1,...,n and edge set ((i,j),(i,j ±1)) ∪ ((1,i),(2,i+σ)) | σ = -⌊(s-1)/2⌋...,0,...,⌊s/2⌋. In this paper we determine...

On the total k-domination number of graphs

Adel P. Kazemi (2012)

Discussiones Mathematicae Graph Theory

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

Éric Sopena (2012)

Discussiones Mathematicae Graph Theory

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The oriented chromatic number of an oriented graph $^{\to} G$ is the minimum order of an oriented graph $^{\to} H$ such that $^{\to} G$ admits a homomorphism to $^{\to} H$ . The oriented chromatic number of an undirected graph G is then the greatest oriented chromatic number of its orientations. In this paper, we introduce the new notion of the upper oriented chromatic number of an undirected graph G, defined as the minimum order of an oriented graph $^{\to} U$ such that every orientation $^{\to} G$ of G admits a homomorphism to $^{\to} U$ . We give...

Graphs with large double domination numbers

Michael A. Henning (2005)

Discussiones Mathematicae Graph Theory

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In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V(G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number $γ_{\times 2} (G)$ . If G ≠ C₅ is a connected graph of order n with minimum degree at least 2, then we show that $γ_{\times 2} (G) \leq 3 n / 4$ and we characterize those graphs achieving equality.

On characterization of uniquely 3-list colorable complete multipartite graphs

Yancai Zhao, Erfang Shan (2010)

Discussiones Mathematicae Graph Theory

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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...

The hull number of strong product graphs

A.P. Santhakumaran, S.V. Ullas Chandran (2011)

Discussiones Mathematicae Graph Theory

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For a connected graph G with at least two vertices and S a subset of vertices, the convex hull ${[S]}_{G}$ is the smallest convex set containing S. The hull number h(G) is the minimum cardinality among the subsets S of V(G) with ${[S]}_{G} = V (G)$ . Upper bound for the hull number of strong product G ⊠ H of two graphs G and H is obtainted. Improved upper bounds are obtained for some class of strong product graphs. Exact values for the hull number of some special classes of strong product graphs are obtained. Graphs...

Rotation and jump distances between graphs

Gary Chartrand, Heather Gavlas, Héctor Hevia, Mark A. Johnson (1997)

Discussiones Mathematicae Graph Theory

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A graph H is obtained from a graph G by an edge rotation if G contains three distinct vertices u,v, and w such that uv ∈ E(G), uw ∉ E(G), and H = G-uv+uw. A graph H is obtained from a graph G by an edge jump if G contains four distinct vertices u,v,w, and x such that uv ∈ E(G), wx∉ E(G), and H = G-uv+wx. If a graph H is obtained from a graph G by a sequence of edge jumps, then G is said to be j-transformed into H. It is shown that for every two graphs G and H of the same order (at least...

Power Domination in Knödel Graphs and Hanoi Graphs

Seethu Varghese, A. Vijayakumar, Andreas M. Hinz (2018)

Discussiones Mathematicae Graph Theory

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In this paper, we study the power domination problem in Knödel graphs WΔ,2ν and Hanoi graphs [...] Hpn $H_{p}^{n}$ . We determine the power domination number of W3,2ν and provide an upper bound for the power domination number of Wr+1,2r+1 for r ≥ 3. We also compute the k-power domination number and the k-propagation radius of [...] Hp2 $H_{p}^{2}$ .

Edge-connectivity of strong products of graphs

Bostjan Bresar, Simon Spacapan (2007)

Discussiones Mathematicae Graph Theory

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The strong product G₁ ⊠ G₂ of graphs G₁ and G₂ is the graph with V(G₁)×V(G₂) as the vertex set, and two distinct vertices (x₁,x₂) and (y₁,y₂) are adjacent whenever for each i ∈ 1,2 either $x_{i} = y_{i}$ or $x_{i} y_{i} \in E (G_{i})$ . In this note we show that for two connected graphs G₁ and G₂ the edge-connectivity λ (G₁ ⊠ G₂) equals minδ(G₁ ⊠ G₂), λ(G₁)(|V(G₂)| + 2|E(G₂)|), λ(G₂)(|V(G₁)| + 2|E(G₁)|). In addition, we fully describe the structure of possible minimum edge cut sets in strong products of graphs.

On choosability of complete multipartite graphs $K_{4, 3 * t, 2 * (k - 2 t - 2), 1 * (t + 1)}$

Guo-Ping Zheng, Yu-Fa Shen, Zuo-Li Chen, Jin-Feng Lv (2010)

Discussiones Mathematicae Graph Theory

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A graph G is said to be chromatic-choosable if ch(G) = χ(G). Ohba has conjectured that every graph G with 2χ(G)+1 or fewer vertices is chromatic-choosable. It is clear that Ohba’s conjecture is true if and only if it is true for complete multipartite graphs. In this paper we show that Ohba’s conjecture is true for complete multipartite graphs $K_{4, 3 * t, 2 * (k - 2 t - 2), 1 * (t + 1)}$ for all integers t ≥ 1 and k ≥ 2t+2, that is, $c h (K_{4, 3 * t, 2 * (k - 2 t - 2), 1 * (t + 1)}) = k$ , which extends the results $c h (K_{4, 3, 2 * (k - 4), 1 * 2}) = k$ given by Shen et al. (Discrete Math. 308 (2008) 136-143), and $c h (K_{4, 3 * 2, 2 * (k - 6), 1 * 3}) = k$ ...

The order of uniquely partitionable graphs

Izak Broere, Marietjie Frick, Peter Mihók (1997)

Discussiones Mathematicae Graph Theory

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Let ₁,...,ₙ be properties of graphs. A (₁,...,ₙ)-partition of a graph G is a partition V₁,...,Vₙ of V(G) such that, for each i = 1,...,n, the subgraph of G induced by $V_{i}$ has property $_{i}$ . If a graph G has a unique (₁,...,ₙ)-partition we say it is uniquely (₁,...,ₙ)-partitionable. We establish best lower bounds for the order of uniquely (₁,...,ₙ)-partitionable graphs, for various choices of ₁,...,ₙ.

4-cycle properties for characterizing rectagraphs and hypercubes

Khadra Bouanane, Abdelhafid Berrachedi (2017)

Czechoslovak Mathematical Journal

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A $(0, 2)$ -graph is a connected graph, where each pair of vertices has either 0 or 2 common neighbours. These graphs constitute a subclass of $(0, λ)$ -graphs introduced by Mulder in 1979. A rectagraph, well known in diagram geometry, is a triangle-free $(0, 2)$ -graph. $(0, 2)$ -graphs include hypercubes, folded cube graphs and some particular graphs such as icosahedral graph, Shrikhande graph, Klein graph, Gewirtz graph, etc. In this paper, we give some local properties of 4-cycles in $(0, λ)$ -graphs and more specifically...

Total domination of Cartesian products of graphs

Xinmin Hou (2007)

Discussiones Mathematicae Graph Theory

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Let γₜ(G) and $γ_{p r} (G)$ denote the total domination and the paired domination numbers of graph G, respectively, and let G □ H denote the Cartesian product of graphs G and H. In this paper, we show that γₜ(G)γₜ(H) ≤ 5γₜ(G □ H), which improves the known result γₜ(G)γₜ(H) ≤ 6γₜ(G □ H) given by Henning and Rall.

On 𝓕-independence in graphs

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

Discussiones Mathematicae Graph Theory

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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.

Upper bounds for the domination numbers of toroidal queens graphs

Christina M. Mynhardt (2003)

Discussiones Mathematicae Graph Theory

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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.

2-halvable complete 4-partite graphs

Dalibor Fronček (1998)

Discussiones Mathematicae Graph Theory

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A complete 4-partite graph $K_{m ₁, m ₂, m ₃, m ₄}$ is called d-halvable if it can be decomposed into two isomorphic factors of diameter d. In the class of graphs $K_{m ₁, m ₂, m ₃, m ₄}$ with at most one odd part all d-halvable graphs are known. In the class of biregular graphs $K_{m ₁, m ₂, m ₃, m ₄}$ with four odd parts (i.e., the graphs $K_{m, m, m, n}$ and $K_{m, m, n, n}$ ) all d-halvable graphs are known as well, except for the graphs $K_{m, m, n, n}$ when d = 2 and n ≠ m. We prove that such graphs are 2-halvable iff n,m ≥ 3. We also determine a new class of non-halvable graphs $K_{m ₁, m ₂, m ₃, m ₄}$ with three...