Number Of Complete Bipartite Graphs Research Articles

The saturation number of K3,3

A graph G is called F-saturated if G does not contain F as a subgraph (not necessarily induced) but the addition of any missing edge to G creates a copy of F. The saturation number of F, denoted by sat(n,F), is the minimum number of edges in an n-vertex F-saturated graph. Determining the saturation number of complete bipartite graphs is one of the most important problems in the study of saturation numbers. The value of sat(n,K2,2) was shown to be ⌊3n−52⌋ by Ollmann, and a shorter proof was later given by Tuza. For K2,3, there has been a series of study aiming to determine sat(n,K2,3) over the years. This was finally achieved by Chen who confirmed a conjecture of Bohman, Fonoberova, and Pikhurko that sat(n,K2,3)=2n−3 for all n≥5. Pikhurko and Schmitt conjectured that sat(n,K3,3)=(3+o(1))n. In this paper, for n≥9, we give an upper bound of 3n−9 for sat(n,K3,3), and prove that 3n−9 is also a lower bound when the minimum degree of a K3,3-saturated graph is 2 or 5, where it is trivial when the minimum degree is greater than 5.

Minimizing the number of complete bipartite graphs in a Ks-saturated graph

A graph $G$ is $F$-saturated if it contains no copy of $F$ as a subgraph but the addition of any new edge to $G$ creates a copy of $F$. We prove that for $s \geq 3$ and $t \geq 2$, the minimum number of copies of $K_{1,t}$ in a $K_s$-saturated graph is $\Theta ( n^{t/2})$. More precise results are obtained when $t = 2$ where the problem is related to Moore graphs with diameter 2 and girth 5. We prove that for $s \geq 4$ and $t \geq 3$, the minimum number of copies of $K_{2,t}$ in an $n$-vertex $K_s$-saturated graph is at least $\Omega( n^{t/5 + 8/5})$ and at most $O(n^{t/2 + 3/2})$. These results answer a question of Chakraborti and Loh. General estimates on the number of copies of $K_{a,b}$ in a $K_s$-saturated graph are also obtained, but finding an asymptotic formula remains open.

Affirmative Solutions on Local Antimagic Chromatic Number

An edge labeling of a connected graph $$G = (V, E)$$ is said to be local antimagic if it is a bijection $$f:E \rightarrow \{1,\ldots ,|E|\}$$ such that for any pair of adjacent vertices x and y, $$f^+(x)\not = f^+(y)$$ , where the induced vertex label $$f^+(x)= \sum f(e)$$ , with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by $$\chi _{la}(G)$$ , is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, we give counterexamples to the lower bound of $$\chi _{la}(G \vee O_2)$$ that was obtained in [Local antimagic vertex coloring of a graph, Graphs Combin. 33:275–285 (2017)]. A sharp lower bound of $$\chi _{la}(G\vee O_n)$$ and sufficient conditions for the given lower bound to be attained are obtained. Moreover, we settled Theorem 2.15 and solved Problem 3.3 in the affirmative. We also completely determined the local antimagic chromatic number of complete bipartite graphs.

Closing in on Hill's Conjecture

Borrowing L\'aszl\'o Sz\'ekely's lively expression, we show that Hill's conjecture is "asymptotically at least 98.5% true". This long-standing conjecture states that the crossing number cr($K_n$) of the complete graph $K_n$ is $H(n) := \frac{1}{4}\lfloor \frac{n}{2}\rfloor \lfloor \frac{n-1}{2}\rfloor \lfloor \frac{n-2}{2}\rfloor \lfloor\frac{n-3}{2}\rfloor$, for all $n\ge 3$. This has been verified only for $n\le 12$. Using flag algebras, Norin and Zwols obtained the best known asymptotic lower bound for the crossing number of complete bipartite graphs, from which it follows that for every sufficiently large $n$, cr$(K_n) > 0.905\, H(n)$. Also using flag algebras, we prove that asymptotically cr$(K_n)$ is at least $0.985\, H(n)$. We also show that the spherical geodesic crossing number of $K_n$ is asymptotically at least $0.996\, H(n)$.

The Fast Search Number of a Complete k-Partite Graph

Research on graph searching has recently gained interest in computer science, mathematics, and physics. This paper studies fast searching of a fugitive in a graph, a model that was introduced by Dyer et al. (in: Fleischer, Xu (eds.) Algorithmic aspects in information and management, Springer, New York, 2008). We provide lower bounds and upper bounds on the fast search number (i.e., the minimum number of searchers required for capturing the fugitive) of complete k-partite graphs. We also investigate some special classes of complete k-partite graphs, such as complete bipartite graphs and complete split graphs. We solve the open problem of determining the fast search number of complete bipartite graphs, and present upper and lower bounds on the fast search number of complete split graphs.

Strong Geodetic Number of Complete Bipartite Graphs and of Graphs with Specified Diameter

The strong geodetic problem is a recent variation of the classical geodetic problem. For a graph G, its strong geodetic number $${{\mathrm{sg}}}(G)$$ is the cardinality of a smallest vertex subset S, such that each vertex of G lies on one fixed shortest path between a pair of vertices from S. In this paper, some general properties of the strong geodetic problem are studied, especially in connection with the diameter of a graph. The problem is also solved for balanced complete bipartite graphs.

Three topics in online list coloring

In online list coloring (introduced by Zhu and by Schauz in 2009), on each round the set of vertices having a particular color in their lists is revealed, and the coloring algorithm chooses an independent subset to receive that color. The paint number of a graph G is the least k such that there is an algorithm to produce a successful coloring with no vertex being shown more than k times; it is at least the choice number. We study paintability of joins with complete or empty graphs, obtaining a partial result toward the paint analogue of Ohba’s Conjecture. We also determine upper and lower bounds on the paint number of complete bipartite graphs and characterize 3-paintcritical graphs.

Reduction of symmetric semidefinite programs using the regular $$\ast$$ -representation

We consider semidefinite programming problems on which a permutation group is acting. We describe a general technique to reduce the size of such problems, exploiting the symmetry. The technique is based on a low-order matrix $$*$$-representation of the commutant (centralizer ring) of the matrix algebra generated by the permutation matrices. We apply it to extending a method of de Klerk et al. that gives a semidefinite programming lower bound to the crossing number of complete bipartite graphs. It implies that cr(K 8,n ) ≥ 2.9299n 2 − 6n, cr(K 9,n ) ≥ 3.8676n 2 − 8n, and (for any m ≥ 9) $$\lim_{n\to\infty}\frac{{\rm cr}(K_{m,n})}{Z(m,n)}\geq 0.8594\frac{m}{m-1},$$ where Z(m,n) is the Zarankiewicz number $$\lfloor\frac{1}{4}(m-1)^2\rfloor\lfloor\frac{1}{4}(n-1)^2\rfloor$$, which is the conjectured value of cr(K m,n). Here the best factor previously known was 0.8303 instead of 0.8594.

Graphs whose minimal rank is two

Let F be a field, F= (V,E) be an undirected graph on n vertices, and let S(F,G) be the set of all symmetric n x n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the eduges of G. For example, if G is a path, S(F,G) consists of the symmetric irreducible tridiagonal matrices. Let mr(F, G) be the minimum rank over all matrices in S(F,G). Then mr(F,G)=1 if and only if G is the union of a clique with at least 2 vertices and an independent set. If F is an infinite field such that char F≠2, then mr(F,G)≤ 2 if and only if the complement of G is the join of a clique and a graph that is the union of at most two cliques and any number of complete bipartite graphs. A similar result is obtained in the case that F is an infinite field with charF=2. Furthermore, in each case, such graphs are characterized as those for which 6 specific graphs do not occur as induced subgraphs. The number of forbiddern subgraphs is reduced to 4 if the graph is connected. Finally, similar criteria is obtained for the minimum rank of a Hermitian matrix to be less than or equal to two. The complement is the join of a clique and a graph that is the union of any number of cliques and any number of complete bipartite graphs. The number of forbidder subgraphs is now 5, or in the connected case, 3. 

Inertia and biclique decompositions of joins of graphs

We characterize the inertia of A+ B for Hermitian matrices A and B when the rank of B is one. We use this to characterize the inertia of a partial join of two graphs. We then provide graph joins G for which the minimum number of complete bipartite graphs needed in a partition of the edge multi-set of G is equal to the maximum of the number of positive and negative eigenvalues of G.

Decompositions ofKm,n into cubes

For a complete bipartite graph to be decomposable into isomorphic cubes, certain conditions on the number of cube and bipartition vertices must hold. We prove these necessary conditions sufficient in some cases. For cubes of fixed dimension d (indeed for d-regular bipartite graphs in general) we show that proving sufficiency can be reduced to decomposing a finite number of complete bipartite graphs. When t = 2d−1 and r is the remainder on dividing t by d, we show Kt,t is decomposable into d-cubes and an r-factor, where if r > 0 this r-factor itself is decomposable into r-cubes. © 1996 John Wiley & Sons, Inc.

Decomposition of graphs and monotone formula size of homogeneous functions

We show that every graph on n nodes can be partitioned by a number of complete bipartite graphs with O(n 2/log n) nodes with no edge belonging to two of them. Since each partition corresponds directly to a monotone formula for the associated quadratic function we obtain an upper bound for the monotone formula size of quadratic functions. Our method extends to uniform hypergraphs with fixed range and thus to homogeneous functions with fixed length of prime implicants. Finally we give an example of a quadratic function where each monotone formula built from arbitrary partitions of the graph (double edges allowed) is not optimal. That means we disprove the single-level-conjecture for formulae.