Balanced Complete Bipartite Graph Research Articles

Minimum degree stability of C2k+1 ${C}_{2k+1}$‐free graphs

AbstractWe consider the minimum degree stability of graphs forbidding odd cycles: What is the tight bound on the minimum degree to guarantee that the structure of a ‐free graph inherits from the extremal graph (a balanced complete bipartite graph)? Andrásfai, Erdős, and Sós showed that if a ‐free graph on vertices has minimum degree greater than , then it is bipartite. Häggkvist showed that for , if a ‐free graph on vertices has minimum degree greater than , then it is bipartite. Häggkvist also pointed out that this result cannot be extended to . In this paper, we give a complete answer for any . We show that if and is an ‐vertex ‐free graph with , then is bipartite, and the bound is tight.

Spectral extremal graphs for the bowtie

Let Fk be the (friendship) graph obtained from k triangles by sharing a common vertex. The Fk-free graphs of order n which attain the maximal spectral radius was firstly characterized by Cioabă, Feng, Tait and Zhang [Electron. J. Combin. 27 (4) (2020)], and later uniquely determined by Zhai, Liu and Xue [Electron. J. Combin. 29 (3) (2022)] under the condition that n is sufficiently large. In this paper, we get rid of the condition on n being sufficiently large if k=2. The graph F2 is also known as the bowtie. We show that the unique n-vertex F2-free spectral extremal graph is the balanced complete bipartite graph adding an edge in the vertex part with smaller size if n≥7, and the condition n≥7 is tight. Our result is a spectral generalization of a theorem of Erdős, Füredi, Gould and Gunderson [J. Combin. Theory Ser. B 64 (1995)], which states that ex(n,F2)=⌊n2/4⌋+1. Moreover, we study the spectral extremal problem for Fk-free graphs with given number of edges. In particular, we show that the unique m-edge F2-free spectral extremal graph is the join of K2 with an independent set of m−12 vertices if m≥8, and the condition m≥8 is tight.

Unavoidable patterns in locally balanced colourings

AbstractWhich patterns must a two-colouring of $K_n$ contain if each vertex has at least $\varepsilon n$ red and $\varepsilon n$ blue neighbours? We show that when $\varepsilon \gt 1/4$ , $K_n$ must contain a complete subgraph on $\Omega (\log n)$ vertices where one of the colours forms a balanced complete bipartite graph.When $\varepsilon \leq 1/4$ , this statement is no longer true, as evidenced by the following colouring $\chi$ of $K_n$ . Divide the vertex set into $4$ parts nearly equal in size as $V_1,V_2,V_3, V_4$ , and let the blue colour class consist of the edges between $(V_1,V_2)$ , $(V_2,V_3)$ , $(V_3,V_4)$ , and the edges contained inside $V_2$ and inside $V_3$ . Surprisingly, we find that this obstruction is unique in the following sense. Any two-colouring of $K_n$ in which each vertex has at least $\varepsilon n$ red and $\varepsilon n$ blue neighbours (with $\varepsilon \gt 0$ ) contains a vertex set $S$ of order $\Omega _{\varepsilon }(\log n)$ on which one colour class forms a balanced complete bipartite graph, or which has the same colouring as $\chi$ .

Effective resistances and spanning trees in the complete bipartite graph plus a matching

In 2019, Ye and Yan computed the effective resistances in the nearly balanced complete bipartite graph Kn,n−pK2(p≤n). Then the result was extended to Km,n−pK2(p≤min{m,n}) very recently. In this paper, we obtain the effective resistances and the number of spanning trees in any complete bipartite graph plus a matching.

Extending Perfect Matchings to Hamiltonian Cycles in Line Graphs

A graph admitting a perfect matching has the Perfect–Matching–Hamiltonian property (for short the PMH–property) if each of its perfect matchings can be extended to a hamiltonian cycle. In this paper we establish some sufficient conditions for a graph $G$ in order to guarantee that its line graph $L(G)$ has the PMH–property. In particular, we prove that this happens when $G$ is (i) a Hamiltonian graph with maximum degree at most 3, (ii) a complete graph, (iii) a balanced complete bipartite graph with at least 100 vertices, or (iv) an arbitrarily traceable graph. Further related questions and open problems are proposed along the paper.

An approximate version of Jackson’s conjecture

AbstractA diregular bipartite tournament is a balanced complete bipartite graph whose edges are oriented so that every vertex has the same in- and out-degree. In 1981 Jackson showed that a diregular bipartite tournament contains a Hamilton cycle, and conjectured that in fact its edge set can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: for everyε > 0 there existsn0such that every diregular bipartite tournament on 2n≥n0vertices contains a collection of (1/2–ε)ncycles of length at least (2–ε)n. Increasing the degree by a small proportion allows us to prove the existence of many Hamilton cycles: for everyc > 1/2 andε > 0 there existsn0such that everycn-regular bipartite digraph on 2n≥n0vertices contains (1−ε)cnedge-disjoint Hamilton cycles.

Strengthening the Murty–Simon conjecture on diameter 2 critical graphs

A graph is diameter-2-critical if its diameter is 2 but the removal of any edge increases the diameter. A well-studied conjecture, known as the Murty–Simon conjecture, states that any diameter-2-critical graph of order n has at most ⌊n2∕4⌋ edges, with equality if and only if G is a balanced complete bipartite graph. Many partial results about this conjecture have been obtained, in particular it is known to hold for all sufficiently large graphs, for all triangle-free graphs, and for all graphs with a dominating edge. In this paper, we discuss ways in which this conjecture can be strengthened. Extending previous conjectures in this direction, we conjecture that, when we exclude the class of complete bipartite graphs and one particular graph, the maximum number of edges of a diameter-2-critical graph is at most ⌊(n−1)2∕4⌋+1. The family of extremal examples is conjectured to consist of certain twin-expansions of the 5-cycle (with the exception of a set of thirteen special small graphs). Our main result is a step towards our conjecture: we show that the Murty–Simon bound is not tight for non-bipartite diameter-2-critical graphs that have a dominating edge, as they have at most ⌊n2∕4⌋−2 edges. Along the way, we give a shorter proof of the Murty–Simon conjecture for this class of graphs, and stronger bounds for more specific cases. We also characterize diameter-2-critical graphs of order n with maximum degree n−2: they form an interesting family of graphs with a dominating edge and 2n−4 edges.

On graphs with a large number of edge-colorings avoiding a rainbow triangle

Inspired by previous work of Balogh (2006), we show that, given r≥5 and n large, the balanced complete bipartite graph Kn∕2,n∕2 is the n-vertex graph that admits the largest number of r-edge-colorings for which there is no triangle whose edges are assigned three distinct colors. Moreover, we extend this result to lower values of n when r≥10, and we provide approximate results for r∈{3,4}.

On Consensus of Star-Composed Networks with an Application of Laplacian Spectrum

In this paper, we mainly study the performance of star-composed networks which can achieve consensus. Specifically, we investigate the convergence speed and robustness of the consensus of the networks, which can be measured by the smallest nonzero eigenvalue λ2 of the Laplacian matrix and the H2 norm of the graph, respectively. In particular, we introduce the notion of the corona of two graphs to construct star-composed networks and apply the Laplacian spectrum to discuss the convergence speed and robustness for the communication network. Finally, the performances of the star-composed networks have been compared, and we find that the network in which the centers construct a balanced complete bipartite graph has the most advantages of performance. Our research would provide a new insight into the combination between the field of consensus study and the theory of graph spectra.

Local colourings and monochromatic partitions in complete bipartite graphs

We show that for any 2-local colouring of the edges of the balanced complete bipartite graph Kn,n, its vertices can be covered with at most 3 disjoint monochromatic paths. And, we can cover almost all vertices of any complete or balanced complete bipartite r-locally coloured graph with O(r2) disjoint monochromatic cycles. We also determine the 2-local bipartite Ramsey number of a path almost exactly: Every 2-local colouring of the edges of Kn,n contains a monochromatic path on n vertices.

Triangle-free graphs with the maximum number of cycles

It is shown that for n≥141, among all triangle-free graphs on n vertices, the balanced complete bipartite graph K⌈n/2⌉,⌊n/2⌋ is the unique triangle-free graph with the maximum number of cycles. Using modified Bessel functions, tight estimates are given for the number of cycles in K⌈n/2⌉,⌊n/2⌋. Also, an upper bound for the number of Hamiltonian cycles in a triangle-free graph is given.

An Exact Turán Result for Tripartite 3-Graphs

Mantel's theorem says that among all triangle-free graphs of a given order the balanced complete bipartite graph is the unique graph of maximum size. We prove an analogue of this result for 3-graphs. Let $K_4^-=\{123,124,134\}$, $F_6=\{123,124,345,156\}$ and $\mathcal{F}=\{K_4^-,F_6\}$: for $n\neq 5$ the unique $\mathcal{F}$-free 3-graph of order $n$ and maximum size is the balanced complete tripartite 3-graph $S_3(n)$ (for $n=5$ it is $C_5^{(3)}=\{123,234,345,145,125\}$). This extends an old result of Bollobás that $S_3(n) $ is the unique 3-graph of maximum size with no copy of $K_4^-=\{123,124,134\}$ or $F_5=\{123,124,345\}$.

On the equitable vertex arboricity of graphs

An equitable -tree-colouring of a graph G is a t-colouring of vertices of G such that the sizes of any two colour classes differ by at most one and the subgraph induced by each colour class is a forest of maximum degree at most k. The strong equitable vertex k-arboricity, denoted by , is the smallest t such that G has an equitable-tree-colouring for every . In this paper, we give upper bounds for when G is a balanced complete bipartite graph and . For some special cases, we determine the exact values. We also prove that: (1) for every planar graph without 4-cycles, 5-cycles and 6-cycles; (2) for every planar graph with neither 3-cycles nor adjacent 4-cycles.

Cycle-maximal triangle-free graphs

We conjecture that the balanced complete bipartite graph K⌊n/2⌋,⌈n/2⌉ contains more cycles than any other n-vertex triangle-free graph, and we make some progress toward proving this. We give equivalent conditions for cycle-maximal triangle-free graphs; show bounds on the numbers of cycles in graphs depending on numbers of vertices and edges, girth, and homomorphisms to small fixed graphs; and use the bounds to show that among regular graphs, the conjecture holds. We also consider graphs that are close to being regular, with the minimum and maximum degrees differing by at most a positive integer k. For k=1, we show that any such counterexamples have n≤91 and are not homomorphic to C5; and for any fixed k there exists a finite upper bound on the number of vertices in a counterexample. Finally, we describe an algorithm for efficiently computing the matrix permanent (a #P-complete problem in general) in a special case used by our bounds.

Partitioning edge-coloured complete graphs into monochromatic cycles and paths

A conjecture of Erdős, Gyárfás, and Pyber says that in any edge-colouring of a complete graph with r colours, it is possible to cover all the vertices with r vertex-disjoint monochromatic cycles. So far, this conjecture has been proven only for r=2. In this paper we show that in fact this conjecture is false for all r⩾3. In contrast to this, we show that in any edge-colouring of a complete graph with three colours, it is possible to cover all the vertices with three vertex-disjoint monochromatic paths, proving a particular case of a conjecture due to Gyárfás. As an intermediate result we show that in any edge-colouring of the complete graph with the colours red and blue, it is possible to cover all the vertices with a red path, and a disjoint blue balanced complete bipartite graph.

Multidecompositions of the Balanced Complete Bipartite Graph into Paths and Stars

Let and denote a path and a star with edges, respectively. For graphs , , and , a -multidecomposition of is a partition of the edge set of into copies of and copies of with at least one copy of and at least one copy of . In this paper, necessary and sufficient conditions for the existence of the (, )-multidecomposition of the balanced complete bipartite graph are given.

Partitioning 3-coloured complete graphs into three monochromatic paths

In this paper we show that in any edge-colouring of the complete graph by three colours, it is possible to cover all the vertices by three disjoint monochromatic paths. This solves a particular case of a conjecture of Gyarfas. As an intermediate result, we show that in any edge colouring of the complete graph by two colours, it is possible to cover all the vertices by a monochromatic path and a disjoint monochromatic balanced complete bipartite graph.

Turán-type results for partial orders and intersection graphs of convex sets

We prove Ramsey-type results for intersection graphs of geometric objects. In particular, we prove the following bounds, all of which are tight apart from the constant c. There is a constant c > 0 such that for every family F of n convex sets in the plane, the intersection graph of F or its complement contains a balanced complete bipartite graph of size at least cn. There is a constant c > 0 such that for every family F of n x-monotone curves in the plane, the intersection graph G of F contains a balanced complete bipartite graph of size at least cn/log n or the complement of G contains a balanced complete bipartite graph of size at least cn. Our bounds rely on new Turan-type results on incomparability graphs of partially ordered sets.

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 .

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.