Handbook of Product Graphs

Hammack, Richard H.; Imrich, Wilfried; Klavžar, Sandi

doi:10.1201/b10959

Cited by 799 publications

(821 citation statements)

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“…Product graphs are considered in order to gain global information from the factor graphs [6]. Many interesting interconnection networks are based on Cartesian product graphs with simple factors, such as paths and cycles.…”

Section: Introductionmentioning

confidence: 99%

Modeling the packing coloring problem of graphs

Shao

Vesel

2015

Applied Mathematical Modelling

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a b s t r a c tThe frequency assignment problem asks for assigning frequencies to transmitters in a wireless network and includes a variety of specific subproblems. In particular, the packing coloring problem comes from the regulations concerning the assignment of broadcast frequencies to radio stations. The problem is placed on a graph-theoretical footing and modeled as the packing chromatic number v q ðGÞ of a graph G. The packing chromatic number of G is the smallest integer k such that the vertex set VðGÞ can be partitioned into disjoint classes X 1 ; . . . ; X k , with the condition that vertices in X i have pairwise distance greater than i. The determination of the packing chromatic number is known to be NP-hard. This paper presents an integer linear programming model and a satisfiability test model for the packing coloring problem. The proposed models were applied for studying the packing chromatic numbers of Cartesian products of paths and cycles and for the packing chromatic numbers of some distance graphs.

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Section: Introductionmentioning

confidence: 99%

Modeling the packing coloring problem of graphs

Shao

Vesel

2015

Applied Mathematical Modelling

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“…Indeed, G × H need not be connected, even if both factors are. This happens exactly when both factors are bipartite (and connected) and in this case there are exactly two components (see [31] or [15]). Also the following distance formula (see [22]),…”

Section: Preliminariesmentioning

confidence: 96%

“…If e = (g, h)(g ′ , h ′ ) ∈ E(G × H), let p G (e) = gg ′ and p H (e) = hh ′ be the projection of edge e over G and H, respectively. The direct product is associative (see [15]) and hence we can write more factors without brackets: G 1 × · · · × G k = × k i=1 G i . The direct product seems to be the most elusive product among all four standard products (Cartesian, strong, direct and lexicographic).…”

Section: Preliminariesmentioning

confidence: 99%

The b-chromatic index of direct product of graphs

Koch

Peterin

2015

Discrete Applied Mathematics

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a b s t r a c tThe b-chromatic index ϕ ′ (G) of a graph G is the largest integer k such that G admits a proper k-edge coloring in which every color class contains at least one edge incident to edges in every other color class. We give in this work bounds for the b-chromatic index of the direct product of graphs and provide general results for many direct products of regular graphs. In addition, we introduce an integer linear programming model for the b-edge coloring problem, which we use for computing exact results for the direct product of some special graph classes.

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“…The Cartesian and tensor products of G and H are two graphs with the same vertex set V (G) × V (H ). In the Cartesian product, two vertices (x, u) and (y, v) are adjacent if and only if [x = y and uv ∈ E(H )] or [u = v and xy ∈ E(G)].For tensor product, (x, u)(y, v) ∈ E(G H ) if and only if xy ∈ E(G) and uv ∈ E(H ), see[12] for details.…”

mentioning

confidence: 97%