Packing and Covering Triangles in Bilaterally-Complete Tripartite Graphs (original) (raw)
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.
Triangle-free partial graphs and edge covering theorems
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
Combinatorica, 1992
G and H, two simple graphs, can be packed if G is isomorphic to a subgraph of H, the complement of H. A theorem of Catlin, Spencer and Sauer gives a sufficient condition for the existence of packing in terms of the product of the maximal degrees of G and H. We improve this theorem for bipartite graphs. Our condition involves products of a maximum degree with an average degree. Our relaxed condition still guarantees a packing of the two bipartite graphs.
Packing triangles in low degree graphs and indifference graphs
2008
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.
Packing Triangles in Weighted Graphs
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.
Cliques and extended triangles. A necessary condition for planar clique graphs
Discrete Applied Mathematics, 2004
We consider simple, finite and undirected graphs. Given a graph G. I'(G) denotes its vertex set and n = | l'(G). A complete of G is a subset of E(G) inducing a complete subgraph. A clique is a maximal complete. We also use the terms complete and clique to refer to the corresponding subgraphs. A complete C covers the edge uv if the end vertices, u and r, belong to C. A complete edge cover of G is a family of completes covering all its edges. Given .A = a family of nonempty sets, the sets F, are called members of the family. E is pairwise intersecting if the intersection of any two members is not the 1 Researcher partially supported by FOMEC. E-mail addresses: lilianawjmate.unlp.edu.ar (L. Alcon), marisate'mate.unlp.edu.ar (M. Gutierrez).
On the Sizes of Bipartite 1-Planar Graphs
Electron. J. Comb., 2021
A graph is called 111-planar if it admits a drawing in the plane such that each edge is crossed at most once. Let GGG be a bipartite 1-planar graph with nnn ($\ge 4$) vertices and mmm edges. Karpov showed that mle3n−8m\le 3n-8mle3n−8 holds for even nge8n\ge 8nge8 and mle3n−9m\le 3n-9mle3n−9 holds for odd nge7n\ge 7nge7. Czap, Przybylo and uSkrabulakova proved that if the partite sets of GGG are of sizes xxx and yyy, then mle2n+6x−12m\le 2n+6x-12mle2n+6x−12 holds for 2leqxleqy2\leq x\leq y2leqxleqy, and conjectured that mle2n+4x−12m\le 2n+4x-12mle2n+4x−12 holds for xge3x\ge 3xge3 and yge6x−12y\ge 6x-12yge6x−12. In this paper, we settle their conjecture and our result is even under a weaker condition 2lexley2\le x\le y2lexley.
2 S ep 2 01 9 Packing colorings of subcubic outerplanar graphs
2019
Given a graph G and a nondecreasing sequence S = (s1, . . . , sk) of positive integers, the mapping c : V (G) −→ {1, . . . , k} is called an Spacking coloring of G if for any two distinct vertices x and y in c(i), the distance between x and y is greater than si. The smallest integer k such that there exists a (1, 2, . . . , k)-packing coloring of a graph G is called the packing chromatic number of G, denoted χρ(G). The question of boundedness of the packing chromatic number in the class of subcubic (planar) graphs was investigated in several earlier papers; recently it was established that the invariant is unbounded in the class of all subcubic graphs. In this paper, we prove that the packing chromatic number of any 2-connected bipartite subcubic outerplanar graph is bounded by 7. Furthermore, we prove that every subcubic triangle-free outerplanar graph has a (1, 2, 2, 2)-packing coloring, and that there exists a subcubic outerplanar graph with a triangle that does not admit a (1, 2...
One-sided Coverings of Colored Complete Bipartite Graphs
Algorithms and Combinatorics
Assume that the edges of a complete bipartite graph K(A, B) are colored with r colors. In this paper we study coverings of B by vertex disjoint monochromatic cycles, connected matchings, and connected subgraphs. These problems occur in several applications.
A list version of graph packing
Discrete Mathematics, 2016
We consider the following generalization of graph packing. Let G 1 = (V 1 , E 1) and G 2 = (V 2 , E 2) be graphs of order n and G 3 = (V 1 ∪V 2 , E 3) a bipartite graph. A bijection f from V 1 onto V 2 is a list packing of the triple (G 1 , G 2 , G 3) if uv ∈ E 2 implies f (u)f (v) / ∈ E 2 and vf (v) / ∈ E 3 for all v ∈ V 1. We extend the classical results of Sauer and Spencer and Bollobás and Eldridge on packing of graphs with small sizes or maximum degrees to the setting of list packing. In particular, we extend the well-known Bollobás-Eldridge Theorem, proving that if ∆(G 1) ≤ n−2, ∆(G 2) ≤ n−2, ∆(G 3) ≤ n−1, and |E 1 |+|E 2 |+|E 3 | ≤ 2n−3, then either (G 1 , G 2 , G 3) packs or is one of 7 possible exceptions. Hopefully, the concept of list packing will help to solve some problems on ordinary graph packing, as the concept of list coloring did for ordinary coloring.