Complete solution for the rainbow numbers of matchings

For a given graph H and a positive n, the rainbow number ofH, denoted by rb(n,H), is the minimum integer k so that in any edge-coloring of K_n with k colors there is a copy of H whose edges have distinct colors. In 2004, Schiermeyer determined rb(n,kK₂) for all n≥3k+3. The case for smaller values of n (namely, ) remained generally open. In this paper we extend Schiermeyer’s result to all plausible n and hence determine the rainbow number of matchings.     

Rainbow number of matchings in planar graphs

The rainbow number$rb(G,H)$ for the graph $H$ in $G$ is defined to be the minimum integer $c$ such that any $c$-edge-coloring of $G$ contains a rainbow $H$. As one of the most important structures in graphs, the rainbow number of matchings has drawn much attention and has been extensively studied. Jendrol et al. initiated the rainbow number of matchings in planar graphs and they obtained bounds for the rainbow number of the matching $k{K}_2$ in the plane triangulations, where the gap between the lower and upper bounds is $O\left(k^3\right)$. In this paper, we show that the rainbow number of the matching $k{K}_2$ in maximal outerplanar graphs of order $n$ is $n+O\left(k\right)$. Using this technique, we show that the rainbow number of the matching $k{K}_2$ in some subfamilies of plane triangulations of order $n$ is $2n+O\left(k\right)$. The gaps between our lower and upper bounds are only $O\left(k\right)$.     

Bipartite rainbow numbers of matchings

Given two graphs G and H, let f(G,H) denote the maximum number c for which there is a way to color the edges of G with c colors such that every subgraph H of G has at least two edges of the same color. Equivalently, any edge-coloring of G with at least rb(G,H)=f(G,H)+1 colors contains a rainbow copy of H, where a rainbow subgraph of an edge-colored graph is such that no two edges of it have the same color. The number rb(G,H) is called the rainbow number ofHwith respect toG, and simply called the bipartite rainbow number ofH if G is the complete bipartite graph K_m,n. Erd?s, Simonovits and Sós showed that rb(K_n,K₃)=n. In 2004, Schiermeyer determined the rainbow numbers rb(K_n,K_k) for all n≥k≥4, and the rainbow numbers rb(K_n,kK₂) for all k≥2 and n≥3k+3. In this paper we will determine the rainbow numbers rb(K_m,n,kK₂) for all k≥1.     

Vertex-disjoint rainbow triangles in edge-colored graphs

Let $G$ be an edge-colored graph of order $n$. The minimum color degree of $G$, denoted by ${\delta }^c\left(G\right)$, is the largest integer $k$ such that for every vertex $v$, there are at least $k$ distinct colors on edges incident to $v$. We say that an edge-colored graph is rainbow if all its edges have different colors. In this paper, we consider vertex-disjoint rainbow triangles in edge-colored graphs. Li (2013) showed that if ${\delta }^c\left(G\right)\ge (n+1)∕2$, then $G$ contains a rainbow triangle and the lower bound is tight. Motivated by this result, we prove that if $n\ge 20$ and ${\delta }^c\left(G\right)\ge (n+2)∕2$, then $G$ contains two vertex-disjoint rainbow triangles. In particular, we conjecture that if ${\delta }^c\left(G\right)\ge (n+k)∕2$, then $G$ contains $k$ vertex-disjoint rainbow triangles. For any integer $k\ge 2$, we show that if $n\ge 16k-12$ and ${\delta }^c\left(G\right)\ge n∕2+k-1$, then $G$ contains $k$ vertex-disjoint rainbow triangles. Moreover, we provide sufficient conditions for the existence of $k$ edge-disjoint rainbow triangles.     

Note on rainbow cycles in edge-colored graphs

Let G be a graph of order n with an edge-coloring c, and let ${\delta }^c(G)$ denote the minimum color-degree of G. A subgraph F of G is called rainbow if all edges of F have pairwise distinct colors. There have been a lot of results on rainbow cycles of edge-colored graphs. In this paper, we show that (i) if ${\delta }^c(G)>\frac{2n?1}{3}$, then every vertex of G is contained in a rainbow triangle; (ii) if ${\delta }^c(G)>\frac{2n?1}{3}$ and $n\ge 13$, then every vertex of G is contained in a rainbow $C_4$; (iii) if G is complete, $n\ge 7k?17$ and ${\delta }^c(G)>\frac{n?1}{2}+k$, then G contains a rainbow cycle of length at least k, where $k\ge 5$.     

Existence of rainbow matchings in properly edge-colored graphs

Let G be a properly edge-colored graph. A rainbow matching of G is a matching in which no two edges have the same color. Let δ denote the minimum degree of G. We show that if |V(G)| > (δ ² + 14δ + 1)/4, then G has a rainbow matching of size δ, which answers a question asked by G. Wang [Electron. J. Combin., 2011, 18: #N162] affirmatively. In addition, we prove that if G is a properly colored bipartite graph with bipartition (X, Y) and max{|X|, |Y|} > (δ ² + 4δ − 4)/4, then G has a rainbow matching of size δ.     

Connected matchings in chordal bipartite graphs

A connected matching in a graph is a collection of edges that are pairwise disjoint but joined by another edge of the graph. Motivated by applications to Hadwiger’s conjecture, Plummer, Stiebitz, and Toft (2003) introduced connected matchings and proved that, given a positive integer $k$, determining whether a graph has a connected matching of size at least $k$ is NP-complete. Cameron (2003) proved that this problem remains NP-complete on bipartite graphs, but can be solved in polynomial-time on chordal graphs. We present a polynomial-time algorithm that finds a maximum connected matching in a chordal bipartite graph. This includes a novel edge-without-vertex-elimination ordering of independent interest. We give several applications of the algorithm, including computing the Hadwiger number of a chordal bipartite graph, solving the unit-time bipartite margin-shop scheduling problem in the case in which the bipartite complement of the precedence graph is chordal bipartite, and determining–in a totally balanced binary matrix–the largest size of a square sub-matrix that is permutation equivalent to a matrix with all zero entries above the main diagonal.     

Maximum matchings in regular graphs

It was conjectured by Mkrtchyan, Petrosyan and Vardanyan that every graph $G$ with $\Delta \left(G\right)?\delta \left(G\right)\le 1$ has a maximum matching $M$ such that any two $M$-unsaturated vertices do not share a neighbor. The results obtained in Mkrtchyan et al. (2010), Petrosyan (2014) and Picouleau (2010) leave the conjecture unknown only for $k$-regular graphs with $4\le k\le 6$. All counterexamples for $k$-regular graphs $(k\ge 7)$ given in Petrosyan (2014) have multiple edges. In this paper, we confirm the conjecture for all $k$-regular simple graphs and also $k$-regular multigraphs with $k\le 4$.     

Roman domination in regular graphs

A Roman domination function on a graph G=(V(G),E(G)) is a function f:V(G)→{0,1,2} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. The weight of a Roman dominating function is the value f(V(G))=∑_u∈V(G)f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. Cockayne et al. [E. J. Cockayne et al. Roman domination in graphs, Discrete Mathematics 278 (2004) 11-22] showed that γ(G)≤γ_R(G)≤2γ(G) and defined a graph G to be Roman if γ_R(G)=2γ(G). In this article, the authors gave several classes of Roman graphs: P_3k,P_3k+2,C_3k,C_3k+2 for k≥1, K_m,n for min{m,n}≠2, and any graph G with γ(G)=1; In this paper, we research on regular Roman graphs and prove that: (1) the circulant graphs and , n⁄≡1 (mod (2k+1)), (n≠2k) are Roman graphs, (2) the generalized Petersen graphs P(n,2k+1)( (mod 4) and ), P(n,1) (n⁄≡2 (mod 4)), P(n,3) ( (mod 4)) and P(11,3) are Roman graphs, and (3) the Cartesian product graphs are Roman graphs.     

A note on rainbow matchings in strongly edge-colored graphs

The Ryser Conjecture which states that there is a transversal of size $n$ in a Latin square of odd order $n$ is equivalent to finding a rainbow matching of size $n$ in a properly edge-colored $K_{n,n}$ using $n$ colors when $n$ is odd. Let $\delta$ be the minimum degree of a graph. Wang proposed a more general question to find a function $f\left(\delta \right)$ such that every properly edge-colored graph of order $f\left(\delta \right)$ contains a rainbow matching of size $\delta$, which currently has the best bound of $f\left(\delta \right)\le 3.5\delta +2$ by Lo. Babu, Chandran and Vaidyanathan investigated Wang’s question under a stronger color condition. A strongly edge-colored graph is a properly edge-colored graph in which every monochromatic subgraph is an induced matching. Wang, Yan and Yu proved that every strongly edge-colored graph of order at least $2\delta +2$ has a rainbow matching of size $\delta$. In this note, we extend this result to graphs of order at least $2\delta +1$.     

A note on a conjecture on maximum matching in almost regular graphs

Mkrtchyan, Petrosyan, and Vardanyan made the following conjecture: Every graph G with Δ(G)−δ(G)≤1 has a maximum matching whose unsaturated vertices do not have a common neighbor. We disprove this conjecture.     

Maximum matchings in random bipartite graphs and the space utilization of Cuckoo Hash tables

We study the the following question in Random Graphs. We are given two disjoint sets L,R with |L| = n and |R| = m. We construct a random graph G by allowing each x∈L to choose d random neighbours in R. The question discussed is as to the size μ(G) of the largest matching in G. When considered in the context of Cuckoo Hashing, one key question is as to when is μ(G) = n whp? We answer this question exactly when d is at least three. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Complete graphs and complete bipartite graphs without rainbow path

Motivated by Ramsey-type questions, we consider edge-colorings of complete graphs and complete bipartite graphs without rainbow path. Given two graphs $G$ and $H$, the $k$-colored Gallai–Ramsey number $g{r}_k(G:H)$ is defined to be the minimum integer $n$ such that $\left(\genfrac{}{}{0ex}{}{n}{2}\right)\ge k$ and for every $N\ge n$, every rainbow $G$-free coloring (using all $k$ colors) of the complete graph $K_N$ contains a monochromatic copy of $H$. In this paper, we first provide some exact values and bounds of $g{r}_k(P_5:K_t)$. Moreover, we define the $k$-colored bipartite Gallai–Ramsey number $bg{r}_k(G:H)$ as the minimum integer $n$ such that $n^2\ge k$ and for every $N\ge n$, every rainbow $G$-free coloring (using all $k$ colors) of the complete bipartite graph $K_{N,N}$ contains a monochromatic copy of $H$. Furthermore, we describe the structures of complete bipartite graph $K_{n,n}$ with no rainbow $P_4$ and $P_5$, respectively. Finally, we find the exact values of $bg{r}_k(P_4:K_{s,t})$ ($1\le s\le t$), $bg{r}_k(P_4:F)$ (where $F$ is a subgraph of $K_{s,t}$), $bg{r}_k(P_5:K_{1,t})$ and $bg{r}_k(P_5:K_{2,t})$ by using the structural results.