Displaying similar documents to “Total domination in categorical products of graphs”

Packing of graphs

Woźniak Mariusz

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PrefaceThere are two basic reference texts on packing theory: the last chapter of Bollobás's book [6] (1978) and the 4th chapter of Yap's book [85] (1986). They still remain the main references to packing problems. However, many papers related to these problems have recently been published and the reason for writing this survey is to gather in a systematic form results scattered throughout the literature.I wish I could name all who deserve my thanks. I am particularly grateful to A....

Generalizations of the tree packing conjecture

Dániel Gerbner, Balázs Keszegh, Cory Palmer (2012)

Discussiones Mathematicae Graph Theory

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The Gyárfás tree packing conjecture asserts that any set of trees with 2,3,...,k vertices has an (edge-disjoint) packing into the complete graph on k vertices. Gyárfás and Lehel proved that the conjecture holds in some special cases. We address the problem of packing trees into k-chromatic graphs. In particular, we prove that if all but three of the trees are stars then they have a packing into any k-chromatic graph. We also consider several other generalizations of the conjecture. ...

Labeled Embedding Of (n, n-2)-Graphs In Their Complements

M.-A. Tahraoui, E. Duchêne, H. Kheddouci (2017)

Discussiones Mathematicae Graph Theory

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Graph packing generally deals with unlabeled graphs. In [4], the authors have introduced a new variant of the graph packing problem, called the labeled packing of a graph. This problem has recently been studied on trees [M.A. Tahraoui, E. Duchêne and H. Kheddouci, Labeled 2-packings of trees, Discrete Math. 338 (2015) 816-824] and cycles [E. Duchˆene, H. Kheddouci, R.J. Nowakowski and M.A. Tahraoui, Labeled packing of graphs, Australas. J. Combin. 57 (2013) 109-126]. In this note, we...

New Bounds on the Signed Total Domination Number of Graphs

Seyyed Mehdi Hosseini Moghaddam, Doost Ali Mojdeh, Babak Samadi, Lutz Volkmann (2016)

Discussiones Mathematicae Graph Theory

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In this paper, we study the signed total domination number in graphs and present new sharp lower and upper bounds for this parameter. For example by making use of the classic theorem of Turán [8], we present a sharp lower bound on Kr+1-free graphs for r ≥ 2. Applying the concept of total limited packing we bound the signed total domination number of G with δ(G) ≥ 3 from above by [...] . Also, we prove that γst(T) ≤ n − 2(s − s′) for any tree T of order n, with s support vertices and...

Vizing's conjecture and the one-half argument

Bert Hartnell, Douglas F. Rall (1995)

Discussiones Mathematicae Graph Theory

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The domination number of a graph G is the smallest order, γ(G), of a dominating set for G. A conjecture of V. G. Vizing [5] states that for every pair of graphs G and H, γ(G☐H) ≥ γ(G)γ(H), where G☐H denotes the Cartesian product of G and H. We show that if the vertex set of G can be partitioned in a certain way then the above inequality holds for every graph H. The class of graphs G which have this type of partitioning includes those whose 2-packing number is no smaller than γ(G)-1 as...

A Note on Uniquely Embeddable Forests

Justyna Otfinowska, Mariusz Woźniak (2013)

Discussiones Mathematicae Graph Theory

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Let F be a forest of order n. It is well known that if F 6= Sn, a star of order n, then there exists an embedding of F into its complement F. In this note we consider a problem concerning the uniqueness of such an embedding.

Packing Trees Into n-Chromatic Graphs

András Gyárfás (2014)

Discussiones Mathematicae Graph Theory

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We show that if a sequence of trees T1, T2, ..., Tn−1 can be packed into Kn then they can be also packed into any n-chromatic graph.

A note on careful packing of a graph

M. Woźniak (1995)

Discussiones Mathematicae Graph Theory

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Let G be a simple graph of order n and size e(G). It is well known that if e(G) ≤ n-2, then there is an edge-disjoint placement of two copies of G into Kₙ. We prove that with the same condition on size of G we have actually (with few exceptions) a careful packing of G, that is an edge-disjoint placement of two copies of G into Kₙ∖Cₙ.

On dominating the Cartesian product of a graph and K₂

Bert L. Hartnell, Douglas F. Rall (2004)

Discussiones Mathematicae Graph Theory

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In this paper we consider the Cartesian product of an arbitrary graph and a complete graph of order two. Although an upper and lower bound for the domination number of this product follow easily from known results, we are interested in the graphs that actually attain these bounds. In each case, we provide an infinite class of graphs to show that the bound is sharp. The graphs that achieve the lower bound are of particular interest given the special nature of their dominating sets and...

Packing Coloring of Some Undirected and Oriented Coronae Graphs

Daouya Laïche, Isma Bouchemakh, Éric Sopena (2017)

Discussiones Mathematicae Graph Theory

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The packing chromatic number χρ(G) of a graph G is the smallest integer k such that its set of vertices V(G) can be partitioned into k disjoint subsets V1, . . . , Vk, in such a way that every two distinct vertices in Vi are at distance greater than i in G for every i, 1 ≤ i ≤ k. For a given integer p ≥ 1, the p-corona of a graph G is the graph obtained from G by adding p degree-one neighbors to every vertex of G. In this paper, we determine the packing chromatic number of p-coronae...