Fractional arboricity, strength, and principal partitions in graphs and matroids

Catlin, Paul A.; Grossman, Jerrold W.; Hobbs, Arthur M.; Lai, Hong‐Jian

doi:10.1016/0166-218x(92)90002-r

Cited by 68 publications

(56 citation statements)

References 11 publications

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“…Note that the definition of density (and therefore uniform density) given here agrees with the definition given in [13], but differs slightly from the one given in [27]. However, these definitions agree on connected graphs.…”

Section: E(h)|supporting

confidence: 58%

More results on r-inflated graphs: Arboricity, thickness, chromatic number and fractional chromatic number

2010

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The r-inflation of a graph G is the lexicographic product G with K r . A graph is said to have thickness t if its edges can be partitioned into t sets, each of which induces a planar graph, and t is smallest possible. In the setting of the r-inflation of planar graphs, we investigate the generalization of Ringel's famous Earth-Moon problem: What is the largest chromatic number of any thickness-t graph? In particular, we study classes of planar graphs for which we can determine both the thickness and chromatic number of their 2-inflations, and provide bounds on these parameters for their r-inflations. Moreover, in the same setting, we investigate arboricity and fractional chromatic number as well. We end with a list of open questions.

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Section: E(h)|supporting

confidence: 58%

More results on r-inflated graphs: Arboricity, thickness, chromatic number and fractional chromatic number

2010

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“…It was called the strength of G by Cunningham in [3] and extended to matroids. A unified approach to strength and principal partitions in matroids is given by Catlin et al in [2]. For integer values of q, inductive constructions for q-strong and q-superstrong graphs have been given by Nash-Williams [15] and Frank and Király [5], respectively.…”

Section: Further Remarksmentioning

confidence: 99%

Brick partitions of graphs

Jackson

Jordán

2010

Discrete Mathematics

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a b s t r a c tFor each rational number q = b/c where b ≥ c are positive integers, we define a q-brick of G to be a maximal subgraph H of G such that cH has b edge-disjoint spanning trees, and a q-superbrick of G to be a maximal subgraph H of G such that cH − e has b edgedisjoint spanning trees for all edges e of cH, where cH denotes the graph obtained from H by replacing each edge by c parallel edges. We show that the vertex sets of the q-bricks of G partition the vertex set of G, and that the vertex sets of the q-superbricks of G form a refinement of this partition. The special cases when q = 1 are the partitions given by the connected components and the 2-edge-connected components of G, respectively. We obtain structural results on these partitions and describe their relationship to the principal partitions of a matroid.

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“…The 1-balanced graphs and matroids have been studied by many researchers; see [2,7,13,14,16,19,20], and the references listed in those papers. Other names for a 1-balanced graph include ''molecular graph'' [13,14], ''strongly balanced graph'' [12,19], and ''uniformly dense'' [10,11].…”

Section: Introductionmentioning

confidence: 99%

Balanced and 1-balanced graph constructions

Hobbs

Kannan

Lai

et al. 2010

Discrete Applied Mathematics

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a b s t r a c tThere are several density functions for graphs which have found use in various applications.In this paper, we examine two of them, the first being given by b(G) = |E(G)|/|V (G)|, and the other being given by g(G) = |E(G)|/(|V (G)| − ω(G)), where ω(G) denotes the number of components of G. Graphs for which b(H) ≤ b(G) for all subgraphs H of G are called balanced graphs, and graphs for which g(H) ≤ g(G) for all subgraphs H of G are called 1-balanced graphs (also sometimes called strongly balanced or uniformly dense in the literature). Although the functions b and g are very similar, they distinguish classes of graphs sufficiently differently that b(G) is useful in studying random graphs, g(G) has been useful in designing networks with reduced vulnerability to attack and in studying the World Wide Web, and a similar function is useful in the study of rigidity. First we give a new characterization of balanced graphs. Then we introduce a graph construction which generalizes the Cartesian product of graphs to produce what we call a generalized Cartesian product. We show that generalized Cartesian product derived from a tree and 1-balanced graphs are 1-balanced, and we use this to prove that the generalized Cartesian products derived from 1-balanced graphs are 1-balanced.

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Fractional arboricity, strength, and principal partitions in graphs and matroids

Cited by 68 publications

References 11 publications

More results on r-inflated graphs: Arboricity, thickness, chromatic number and fractional chromatic number

More results on r-inflated graphs: Arboricity, thickness, chromatic number and fractional chromatic number

Brick partitions of graphs

Balanced and 1-balanced graph constructions

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