Old and new generalizations of line graphs

Bagga, Jay S.

doi:10.1155/s0161171204310094

Cited by 17 publications

(14 citation statements)

References 11 publications

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“…[15] where some related results are presented. A few other observations are mentioned in the survey [3].…”

Section: Triangle Graphs 3-colorings Hypergraphsmentioning

confidence: 97%

“…Let 1 be induced by the vertices u 1 , u 2 and u 3 and 2 be induced by u 1 , u 2 and u 3 in G. If the common edge shared by 1 and 3 were different from u 1 u 2 then, since 2 and 3 are adjacent in T (G), 3 should be either (u 1 , u 3 , u 3 ) or (u 2 , u 3 , u 3 ). But then u 1 , u 2 , u 3 , u 3 would induce a K 4 in G. This contradiction proves the claim.…”

Section: Graphs With Odd-hole-free Triangle Graphsmentioning

confidence: 99%

“…Indeed, the wheel W 5 is 4-colorable and hence it admits a homomorphism into K 4 , but its anti-Gallai graph contains an induced cycle of length 5. Also, there are graphs (even without triangles) whose chromatic number is greater than 5, so they are not homomorphic to C 3 10 , but their anti-Gallai graphs are odd-hole-free.…”

Section: Concluding Remarks and Open Problemsmentioning

confidence: 99%

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Small Edge Sets Meeting all Triangles of a Graph

Lakshmanan

Bujtás

Tuza

2011

Graphs and Combinatorics

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It was conjectured in 1981 by the third author that if a graph G does not contain more than t pairwise edge-disjoint triangles, then there exists a set of at most 2t edges that shares an edge with each triangle of G. In this paper, we prove this conjecture for odd-wheel-free graphs and for 'triangle-3-colorable' graphs, where the latter property means that the edges of the graph can be colored with three colors in such a way that each triangle receives three distinct colors on its edges. Among the consequences we obtain that the conjecture holds for every graph with chromatic number at most four. Also, two subclasses of K 4 -free graphs are identified, in which the maximum number of pairwise edge-disjoint triangles is equal to the minimum number of edges covering all triangles. In addition, we prove that the recognition problem of triangle-3-colorable graphs is intractable.

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“…[15] where some related results are presented. A few other observations are mentioned in the survey [3].…”

Section: Triangle Graphs 3-colorings Hypergraphsmentioning

confidence: 97%

Section: Graphs With Odd-hole-free Triangle Graphsmentioning

confidence: 99%

Section: Concluding Remarks and Open Problemsmentioning

confidence: 99%

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Small Edge Sets Meeting all Triangles of a Graph

Lakshmanan

Bujtás

Tuza

2011

Graphs and Combinatorics

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“…Our main results in this section are (sharp) lower and upper-bounds on the hyperbolicity of intersection graphs that have been considered in the literature. These comprise the two wellknown families of line graphs and clique graphs (we refer to [1,29] for surveys), along with biclique graphs that have been introduced more recently as extensions of line graphs. By mixing up the two chains of inequality one obtains 2δ(G) ≤ 2δ(L(G)) + 2 and 2δ(L(G)) ≤ 2δ(G) + 2, whence δ(G) ≤ δ(L(G)) + 1 and δ(L(G)) ≤ δ(G) + 1, as desired.…”

Section: Applications To Intersection Graphsmentioning

confidence: 99%

On the hyperbolicity of bipartite graphs and intersection graphs

Coudert¹,

Ducoffe²

2016

Discrete Applied Mathematics

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International audienceHyperbolicity is a measure of the tree-likeness of a graph from a metric perspective. Recently , it has been used to classify complex networks depending on their underlying geometry. Motivated by a better understanding of the structure of graphs with bounded hyperbolicity, we here investigate on the hyperbolicity of bipartite graphs. More precisely, given a bipartite graph B = (V0 ∪ V1 , E) we prove it is enough to consider any one side Vi of the bipartition of B to obtain a close approximate of its hyperbolicity δ(B) — up to an additive constant 2. We obtain from this result the sharp bounds δ(G) − 1 ≤ δ(L(G)) ≤ δ(G) + 1 and δ(G) − 1 ≤ δ(K(G)) ≤ δ(G) + 1 for every graph G, with L(G) and K(G) being respectively the line graph and the clique graph of G. Finally, promising extensions of our techniques to a broader class of intersection graphs are discussed and illustrated with the case of the biclique graph BK(G), for which we prove (δ(G) − 3)/2 ≤ δ(BK(G)) ≤ (δ(G) + 3)/2

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“…Several variations of the super line graph have been considered. A recent survey of line graphs and their generalizations is given in [1].…”

Section: Introductionmentioning

confidence: 99%

The structure of super line graphs

Bagga

Ferrero²,

Ellis³

2005

8th International Symposium on Parallel Architectures,Algorithms and Networks (ISPAN'05)

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For a given graph G = (V, E) and a positive integer k, the super line graph of index k of G is the graph S k (G) which has for vertices all the k-subsets of E(G), and two vertices S and T are adjacent whenever there exist s ∈ S and t ∈ T such that s and t share a common vertex. In the super line multigraph L k (G) we have an adjacency for each such occurrence.We give a formula to find the adjacency matrix of L k (G). If G is a regular graph, we calculate all the eigenvalues of L k (G) and their multiplicities. From those results we give an upper bound on the number of isolated vertices.

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Old and new generalizations of line graphs

Cited by 17 publications

References 11 publications

Small Edge Sets Meeting all Triangles of a Graph

Small Edge Sets Meeting all Triangles of a Graph

On the hyperbolicity of bipartite graphs and intersection graphs

The structure of super line graphs

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