Packing and Covering Triangles in Bilaterally-Complete Tripartite Graphs (original) (raw)
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The minimum number of triangles covering the edges of a graph
Journal of Graph Theory, 1989
Suppose that a connected graph G has n vertices and rn edges, and each edge is contained in some triangle of G. Bounds are established here on the minimum number tm,,(G) of triangles that cover the edges of G. We prove that [ (n-1)/21 I tmln(G) with equality attained only by 3cactii and by strongly related graphs. We obtain sharp upper bounds: if G is not a 4-clique, then tm,,(G) I L(n-1)'/4J, and tmln(G) I rn + 2-r 2-1 for every rn, rn # 1, 2, and 6. The triangle cover number tmln(G) is also bounded above by T(G) = rnn + c, the cyclomatic number of a graph G of order n with rn edges and c connected components. Here w e give a combinatorial proof for tml,(G) I T(G) and characterize the family of all extremal graphs satisfying equality.
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Discrete Mathematics, 1982
In Section 1 some lower bounds are given for the maximal number of edges of a (p-l)colorable partial graph. Among others we show that a graph on n vertices with m edges has a (p-l)-colorable partiai graph with at least mT,.$(;) edges, where T,,p denotes the so called Turin number. These results are used to obtain upper bounds for special edge covering numbers of graphs. In Section 2 we prove the following theorem: If G is a simple graph and or. is the maximal cardinality of a triangle-free edge set of G, then the edges of G can be covered by p triangles and edges. In Section 3 related questions are examined. We will consider only loopless graphs withlout multiple edg.zs. If G = (X, E) is a graph, then the edge set Fc E together with the spanned vertices define a partial graph of G which will be denoted by the same letter F for simplicity reasons. KP stands for a p-clique (compiete graph on p vertices); K3 will be called a
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We consider the problems of finding the maximum number of vertex-disjoint triangles (VTP) and edge-disjoint triangles (ETP) in a simple graph. Both problems are NP-hard. The algorithm with the best approximation ratio known so far for these problems has ratio 3/2 + ε, a result that follows from a more general algorithm for set packing obtained by Hurkens and Schrijver [On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems, SIAM J. Discrete Math. 2(1) (1989) 68-72]. We present improvements on the approximation ratio for restricted cases of VTP and ETP that are known to be APX-hard: we give an approximation algorithm for VTP on graphs with maximum degree 4 with ratio slightly less than 1.2, and for ETP on graphs with maximum degree 5 with ratio 4/3. We also present an exact linear-time algorithm for VTP on the class of indifference graphs.
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SIAM Journal on Discrete Mathematics, 2014
Tuza conjectured that for every graph G, the maximum size ν of a set of edge-disjoint triangles and minimum size τ of a set of edges meeting all triangles satisfy τ ≤ 2ν. We consider an edge-weighted version of this conjecture, which amounts to packing and covering triangles in multigraphs. Several known results about the original problem are shown to be true in this context, and some are improved. In particular, we answer a question of Krivelevich who proved that τ ≤ 2ν * (where ν * is the fractional version of ν), and asked if this is tight. We prove that τ ≤ 2ν * − 1 √ 6 √ ν * and show that this bound is essentially best possible.
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Let N(n,k) be the minimum number of pairwise edge disjoint monochromatic complete graphs Kk in any 2-coloring of the edges of a Kn. Upper and lower bounds on N(n,k) will be given for k⩾3. For k=3, exact values will be given for n⩽11, and these will be used to give a lower bound for N(n,3).
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Graphs and Combinatorics, 2012
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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