Balanced Bipartite Graph Research Articles

Embedding into Bipartite Graphs

The conjecture of Bollob\'as and Koml\'os, recently proved by B\"ottcher, Schacht, and Taraz [Math. Ann. 343(1), 175--205, 2009], implies that for any $\gamma>0$, every balanced bipartite graph on $2n$ vertices with bounded degree and sublinear bandwidth appears as a subgraph of any $2n$-vertex graph $G$ with minimum degree $(1+\gamma)n$, provided that $n$ is sufficiently large. We show that this threshold can be cut in half to an essentially best-possible minimum degree of $(\frac12+\gamma)n$ when we have the additional structural information of the host graph $G$ being balanced bipartite. This complements results of Zhao [to appear in SIAM J. Discrete Math.], as well as Hladk\'y and Schacht [to appear in SIAM J. Discrete Math.], who determined a corresponding minimum degree threshold for $K_{r,s}$-factors, with $r$ and $s$ fixed. Moreover, it implies that the set of Hamilton cycles of $G$ is a generating system for its cycle space.

Cyclability in bipartite graphs

Let \(G=(X,Y,E)\) be a balanced \(2\)-connected bipartite graph and \(S \subset V(G)\). We will say that \(S\) is cyclable in \(G\) if all vertices of \(S\) belong to a common cycle in \(G\). We give sufficient degree conditions in a balanced bipartite graph \(G\) and a subset \(S \subset V(G)\) for the cyclability of the set \(S\).

Ore and Erdős type conditions for long cycles in balanced bipartite graphs

Graphs and Algorithms We conjecture Ore and Erdős type criteria for a balanced bipartite graph of order 2n to contain a long cycle C(2n-2k), where 0 <= k < n/2. For k = 0, these are the classical hamiltonicity criteria of Moon and Moser. The main two results of the paper assert that our conjectures hold for k = 1 as well.

Edge condition for hamiltonicity in balanced tripartite graphs

A well-known theorem of Entringer and Schmeichel asserts that a balanced bipartite graph of order \(2n\) obtained from the complete balanced bipartite \(K_{n,n}\) by removing at most \(n-2\) edges, is bipancyclic. We prove an analogous result for balanced tripartite graphs: If \(G\) is a balanced tripartite graph of order \(3n\) and size at least \(3n^2-2n+2\), then \(G\) contains cycles of all lengths.

Bi-cycle extendable through a given set in balanced bipartite graphs

Let G = ( X , Y ; E ) be a balanced bipartite graph of order 2 n . The path-cover number p c ( H ) of a graph H is the minimum number of vertex-disjoint paths that use up all the vertices of H . S ⊆ V ( G ) is called a balanced set of G if | S ∩ X | = | S ∩ Y | . In this paper, we will give some sufficient conditions for a balanced bipartite graph G satisfying that for every balanced set S , there is a bi-cycle of every length from | S | + 2 p c ( 〈 S 〉 ) up to 2 n through S .

Regular Spanning Subgraphs of Bipartite Graphs of High Minimum Degree

Let $G$ be a simple balanced bipartite graph on $2n$ vertices, $\delta = \delta(G)/n$, and $\rho_0={\delta + \sqrt{2 \delta -1} \over 2}$. If $\delta \ge 1/2$ then $G$ has a $\lfloor \rho_0 n \rfloor$-regular spanning subgraph. The statement is nearly tight.

The linear 3-arboricity of [formula omitted] and [formula omitted

A linear k-forest is a forest whose components are paths of length at most k. The linear k-arboricity of a graph G, denoted by la k ( G ) , is the least number of linear k-forests needed to decompose G. In this paper, we completely determine la k ( G ) when G is a balanced complete bipartite graph K n , n or a complete graph K n , and k = 3 .

Bipartite graphs with every matching in a cycle

We give sufficient Ore-type conditions for a balanced bipartite graph to contain every matching in a hamiltonian cycle or a cycle not necessarily hamiltonian. Moreover, for the hamiltonian case we prove that the condition is almost best possible.

Best Odds for Finding a Perfect Matching in a Bipartite Graph

We will study the best way to reveal a hidden perfect matching in a balanced bipartite graph by eliminating edges, one by one, in the hope that the eliminated edge is not part of the mystery perfect matching. We will look for the strategy that maximizes the odds of finding the perfect matching without revealing a fixed number of the edges in that perfect matching. For a complete bipartite graph, this is equivalent to finding a mystery permutation via negative guesses with only a fixed number of incorrect negative guesses.

Long cycles in graphs and digraphs

Abstract Consider the following problem (solved by Woodall): given p > | G | 2 , find the minimum size of a graph G guaranteeing the existence of a cycle of length p. We prove that a balanced bipartite graph of order 2n and size greater than n ( n − k − 1 ) + k + 1 contains a cycle of length 2 n − 2 k , where n ⩾ 1 2 k 2 + 3 2 k + 4 . In the paper we also formulate and give a solution to an analogous problem for digraphs.

On 2–Factors with Prescribed Properties in a Bipartite Graph

Liu and Yan gave the degree condition for a balanced bipartite graph G = (V 1, V 2;E) to have k vertex–disjoint quadrilaterals containing any given k independent edges e 1, . . . , e k of G, respectively. They also conjectured that for any k independent edges e 1, . . . , e k of G, G has a 2–factor with k cycles C 1,C 2, . . . , C k with respect to {e 1, e 2, . . . , e k } such that k–1 of them are quadrilaterals. In this paper, we prove this conjecture.

Vertex coverings by coloured induced graphs — Frames and Umbrellas

A graph G homogeneously embeds in a graph H if for every vertex x of G and every vertex y of H there is an induced copy of G in H with x at y. The graph G uniformly embeds in H if for every vertex y of H there is an induced copy of G in H containing y. For positive integer k, let f k(G) (respectively, g k(G)) be the minimum order of a graph H whose edges can be k-coloured such that for each colour, G homogeneously embeds (respectively, uniformly embeds) in the graph given by V(H) and the edges of that colour. We investigate the values f 2(G) and g 2(G) for special classes of G, in particular when G is a star or balanced complete bipartite graph. Then we investigate f k(G) and g k(G) when k ≥ 3 and G is a complete graph.

A note on internally disjoint alternating paths in bipartite graphs

Let G be a balanced bipartite graph with partite sets X and Y, which has a perfect matching, and let x ∈ X and y ∈ Y . Let k be a positive integer. Then we prove that if G has k internally disjoint alternating paths between x and y with respect to some perfect matching, then G has k internally disjoint alternating paths between x and y with respect to every perfect matching.

On $k$-Ordered Bipartite Graphs

In 1997, Ng and Schultz introduced the idea of cycle orderability. For a positive integer $k$, a graph $G$ is k-ordered if for every ordered sequence of $k$ vertices, there is a cycle that encounters the vertices of the sequence in the given order. If the cycle is also a hamiltonian cycle, then $G$ is said to be k-ordered hamiltonian. We give minimum degree conditions and sum of degree conditions for nonadjacent vertices that imply a balanced bipartite graph to be $k$-ordered hamiltonian. For example, let $G$ be a balanced bipartite graph on $2n$ vertices, $n$ sufficiently large. We show that for any positive integer $k$, if the minimum degree of $G$ is at least $(2n+k-1)/4$, then $G$ is $k$-ordered hamiltonian.

Vertex Coverings by Coloured Induced Graphs—Frames and Umbrellas

Abstract A graph G homogeneously embeds in a graph H if for every vertex x of G and every vertex y of H there is an induced copy of G in H with x at y. The graph G uniformly embeds in H if for every vertex y of H there is an induced copy of G in H containing y. For positive integer k, let fk(G) (respectively, gk(G)) be the minimum order of a graph H whose edges can be k-coloured such that for each colour, G homogeneously embeds (respectively, uniformly embeds) in the graph given by V(H) and the edges of that colour. We investigate the values f2(G) and g2(G) for special classes of G, in particular when G is a star or balanced complete bipartite graph. Then we investigate fk(G) and gk(G) when k ≥ 3 and G is a complete graph.

2-factors in dense bipartite graphs

An n- ladder is a balanced bipartite graph with vertex sets A={ a,…, a n } and B={ b 1,…, b n } such that a i ∼ b j iff | i− j|⩽1. We use techniques developed recently by Komlós et al. (1997) to show that if G=( U, V, E) is a bipartite graph with | U|= n=| V|, with n sufficiently large, and the minimum degree of G is at least n/2+1, then G contains an n-ladder. This answers a question of Wang.

The hamiltonicity of bipartite graphs involving neighborhood unions

Let G=(X,Y) be a 2-connected balanced bipartite graph with |X|=|Y|=n. In this paper, we prove that if |N(x 1)∪N(x 2)|+|N(y 1)∪N(y 2)|⩾n+2 for any {x 1,x 2}⊆X and {y 1,y 2}⊆Y , then G is hamiltonian except when G is a special graph on 8 or on 12 vertices.

Cycles in 2-Factors of Balanced Bipartite Graphs

In the study of hamiltonian graphs, many well known results use degree conditions to ensure sufficient edge density for the existence of a hamiltonian cycle. Recently it was shown that the classic degree conditions of Dirac and Ore actually imply far more than the existence of a hamiltonian cycle in a graph G, but also the existence of a 2-factor with exactly k cycles, where \(\). In this paper we continue to study the number of cycles in 2-factors. Here we consider the well-known result of Moon and Moser which implies the existence of a hamiltonian cycle in a balanced bipartite graph of order 2n. We show that a related degree condition also implies the existence of a 2-factor with exactly k cycles in a balanced bipartite graph of order 2n with \(\).

On 2-factors containing 1-factors in bipartite graphs

Abstract Moon and Moser (Israel J. Math. 1 (1962) 163–165) showed that if G is a balanced bipartite graph of order 2 n and minimum degree σ ⩾ ( n + 1)/2, then G is hamiltonian. Recently, it was shown that their well-known degree condition also implies the existence of a 2-factor with exactly k cycles provided n ⩾ max {52, 2 k 2 + 1}. In this paper, we show that a similar degree condition implies that for each perfect matching M , there exists a 2-factor with exactly k cycles including all edges of M .

On 2-factors containing 1-factors in bipartite graphs

Moon and Moser (Israel J. Math. 1 (1962) 163–165) showed that if G is a balanced bipartite graph of order 2 n and minimum degree σ ⩾ ( n + 1)/2, then G is hamiltonian. Recently, it was shown that their well-known degree condition also implies the existence of a 2-factor with exactly k cycles provided n ⩾ max {52, 2 k 2 + 1}. In this paper, we show that a similar degree condition implies that for each perfect matching M, there exists a 2-factor with exactly k cycles including all edges of M.