anti-Ramsey Number Research Articles

Anti-Ramsey number for perfect matchings in 3-regular bipartite graphs

The anti-Ramsey number AR(G,H) is the maximum number of colors in an edge-coloring of a graph in the family G without any rainbow copy of H. The anti-Ramsey number for matchings has been studied extensively in several graph families, while the problem for perfect matchings was only solved in complete graphs. The anti-Ramsey number of matchings in the family of regular bipartite graphs of order large enough was determined by Li and Xu. Jin improved their bounds on the order of 3-regular bipartite graphs. In this paper, we consider the problem for perfect matchings in 3-regular bipartite graphs. Let G be the family of 3-regular bipartite graphs with m vertices in each partite set. First, we characterize the structure of extremal graphs for the Turán number of perfect matchings in graphs with maximum degree three. Using this characterization, we show that AR(G,mK2)=3m−3. Moreover, we show that AR(F,mK2)=3m−5, where F is the subfamily of 3-edge-connected graphs in G. In order to prove this result, we characterize the structure of bipartite graphs, which contains no perfect matchings, with maximum degree three and size Turán number minus one.

Rainbow disjoint union of P4 and a matching in complete graphs

An edge-colored graph G is called rainbow if all the colors on its edges are distinct. Given graphs G and H, the anti-Ramsey number AR(G,H) is the maximum number k such that there exists an edge-coloring of G with exactly k colors containing no rainbow copy of H. Recently, the anti-Ramsey problem for disjoint union of graphs received much attention. In particular, several researchers focused on the problem for graphs consisting of small components. In this paper, we continue the work in this line. We refine the bound of n and determine the precise value of AR(Kn,P4∪tP2) for all n≥2t+4 in complete graphs.

极大外平面图中树的 Anti-Ramsey 数

极大外平面图中树的 Anti-Ramsey 数

Anti-Ramsey number of disjoint union of star-like hypergraphs

Given an r-graph (or r-uniform hypergraph) F, the anti-Ramsey number ar(n,r,F) is the minimum number c of colors such that for any edge-coloring of the complete r-graph Knr on n vertices with at least c colors, there is a subgraph F of Knr whose edges have distinct colors. Let S3(r) be the linear star of size three in r-graphs. In this paper, we obtain the exact anti-Ramsey number ar(n,r,S3(r)) for all r≥3 and sufficiently large n. An r-graph is star-like if all of its edges share at least one common vertex. Moreover, let F be an r-graph which is a vertex-disjoint union of k+1 star-like r-graphs F0,F1,…,Fk, where F0=S3(r) and each Fi (i=1,…,k) contains a subgraph isomorphic F0. We prove that for all r≥3,k≥1 and sufficiently large n, ar(n,r,F)=(nr)−(n−kr)+ar(n−k,r,S3(r)).

Anti-Ramsey number of matchings in outerplanar graphs

A subgraph of an edge-colored graph is rainbow if all of its edges have different colors. Given a graph family G and a graph H, the anti-Ramsey number AR(G,H) is the maximum number of colors in an edge-coloring of a graph G∈G which has no rainbow copy of H. When G={G}, we write AR(G,H) for short. Erdős, Simonovits and Sós introduced these numbers in the 1970s, they studied the case when G is a complete graph. Since then, many results have appeared considering G and H belonging to specific graph classes. In particular, Jendrol’, Schiermeyer and Tu (Jendrol’ et al., 2014) obtained an upper bound for AR(Tn,kK2) in terms of n and k for k≥3 and n≥2k, where Tn is the family of maximal planar graphs on n vertices. Jin and Ye (Jin and Ye, 2018) continued the study of anti-Ramsey numbers of matchings in planar graphs, by deriving an upper bound for AR(Mn,kK2), where Mn is the family of maximal outerplanar graphs on n vertices. In both works, examples are shown for which there are colorings using many colors while avoiding rainbow matchings, but the numbers of colors in these examples do not reach the upper bounds. The bounds of AR(Tn,kK2) were improved in following works by other researchers. This paper goes in the same direction of these two works, proving exact value for AR(Bn,kK2) in terms of n and k for n≥6 and k≥3, where Bn is the family of maximal bipartite outerplanar graphs on n vertices which contains a Hamilton cycle.

Anti-Ramsey Number of Matchings in 3-Uniform Hypergraphs

Anti-Ramsey Number of Matchings in 3-Uniform Hypergraphs

Anti-Ramsey problems in the Mycielskian of a cycle

Let G and H be two graphs. The maximum integer k, for which there exists an edge coloring ϕ:E(G)→{1,2,…,k} that makes every copy of H has at least two edges with the same color, is the anti-Ramsey number of G with respect to H. Mycielski developed an interesting graph transformation that transforms G into the Mycielskian μ(G) of G. In this paper, we determine the anti-Ramsey number of μ(Cn) with respect to cycles of length 4, 2n and 2n+1, respectively.

Computing the anti-Ramsey number for trees in complete tripartite graph

The anti-Ramsey number AR(K,F) is the largest color number in an edge-coloring of K that contains no rainbow copies of graphs in the family F. In this paper, we determine the exact value of the anti-Ramsey number for trees in a balanced complete tripartite graph.

Turán Numbers and Anti-Ramsey Numbers for Short Cycles in Complete $3$-Partite Graphs

We call a $4$-cycle in $K_{n_{1}, n_{2}, n_{3}}$ multipartite (denoted by $C_{4}^{\text{multi}}$) if it contains at least one vertex in each part of $K_{n_{1}, n_{2}, n_{3}}$ . The Turán number of $ C_{4}^{\text{multi}})$ and $\{C_{3}, C_{4}^{\text{multi}}\})$ is determined in the paper as well as the anti-Ramsey number $\text{ar}(K_{n_{1},n_{2},n_{3}}, C_{4}^{\text{multi}})$. We prove that $\text{ex}(K_{n_{1},n_{2},n_{3}}, C_{4}^{\text{multi}})=n_{1}n_{2}+2n_{3}$ and$\text{ar}(K_{n_{1},n_{2},n_{3}}, C_{4}^{\text{multi}})=\text{ex}(K_{n_{1},n_{2},n_{3}}, \{C_{3}, C_{4}^{\text{multi}}\})+1=n_{1}n_{2}+n_{3}+1,$ where $n_{1}\ge n_{2}\ge n_{3}\ge 1.$

Anti-Ramsey Numbers of Cycles of Length Three in Uniform Hypergraphs

Anti-Ramsey Numbers of Cycles of Length Three in Uniform Hypergraphs

Anti-Ramsey numbers for cycles in n-prisms

An edge colored graph is called rainbow if all the colors on its edges are distinct. A rainbow copy of a graph H in an edge colored graph G is a subgraph of G isomorphic to H such that the coloring restricted to H is rainbow. Let G and H be two graphs. The anti-Ramsey number A r ( G , H ) is the maximum number of colors in an edge coloring of G which has no rainbow copy of H . For n ≥ 3 , the n -prism is the cartesian product C n □ K 2 . In this paper, we determine the anti-Ramsey numbers for cycles in n -prisms.

Improved approximation for maximum edge colouring problem

The anti-Ramsey number , a r ( G , H ) is the minimum integer k such that in any edge colouring of G with k colours there is a rainbow subgraph isomorphic to H , namely, a copy of H with each of its edges assigned a different colour. The notion was introduced by Erdös and Simonovits in 1973. Since then the parameter has been studied extensively. The case when H is a star graph was considered by several graph theorists from the combinatorial point of view. Recently this case received the attention of researchers from the algorithm community because of its applications in interface modelling of wireless networks. To the algorithm community, the problem is known as maximum edge q -colouring problem: Find a colouring of the edges of G , maximizing the number of colours satisfying the constraint that each vertex spans at most q colours on its incident edges. It is easy to see that the maximum value of the above optimization problem equals a r ( G , K 1 , q + 1 ) − 1 . In this paper, we study the maximum edge 2-colouring problem from the approximation algorithm point of view. The case q = 2 is particularly interesting due to its application in real-life problems. Algorithmically, this problem is known to be NP-hard for q ≥ 2 . For the case of q = 2 , it is also known that no polynomial-time algorithm can approximate to a factor less than 3 / 2 assuming the unique games conjecture. Feng et al. showed a 2-approximation algorithm for this problem. Later Adamaszek and Popa presented a 5 / 3 -approximation algorithm with the additional assumption that the input graph has a perfect matching. Note that the obvious but the only known algorithm issues different colours to the edges of a maximum matching (say M ) and different colours to the connected components of G ∖ M . In this article, we give a new analysis of the aforementioned algorithm to show that for triangle-free graphs with perfect matching the approximation ratio is 8 / 5 . We also show that this algorithm cannot achieve a factor better than 58 / 37 on triangle free graphs that has a perfect matching. The contribution of the paper is a completely new, deeper and closer analysis of how the optimum achieves a higher number of colours than the matching based algorithm, mentioned above.

Anti-Ramsey numbers for vertex-disjoint triangles

An edge-colored graph is called rainbow if all the colors on its edges are distinct. Given a positive integer n and a graph G, the anti-Ramsey number ar(n,G) is the maximum number of colors in an edge-coloring of Kn with no rainbow copy of G. Denote by kC3 the union of k vertex-disjoint copies of C3. In this paper, we determine the anti-Ramsey number ar(n,kC3) for n=3k and n≥2k2−k+2, respectively. When 3k≤n≤2k2−k+2, we give lower and upper bounds for ar(n,kC3).

Anti-Ramsey numbers for trees in complete multi-partite graphs

Let G be a complete multi-partite graph of order n. In this paper, we consider the anti-Ramsey number ar(G,Tq) with respect to G and the set Tq of trees with q edges, where 2≤q≤n−1. For the case q=n−1, the result has been obtained by Lu, Meier and Wang. We will extend it to q<n−1. We first show that ar(G,Tq)=ℓq(G)+1, where ℓq(G) is the maximum size of a disconnected spanning subgraph H of G with the property that any two components of H together have at most q vertices. Using this equality, we obtain the exact values of ar(G,Tq) for n−3≤q≤n−1. Moreover, for the general case when (4n−2)/5≤q≤n−1, ar(G,Tq) can be determined by a simple algorithm. In particular, the explicit expression of ar(G,Tq) is given when G has a partite set much larger than all the other partite sets.

The Outer-Planar Anti-Ramsey Number of Matchings

A subgraph H of an edge-colored graph G is called rainbow if all of its edges have different colors. Let ar(G,H) denote the maximum positive integer t, such that there is a t-edge-colored graph G without any rainbow subgraph H. We denote by kK2 a matching of size k and On the class of all maximal outer-planar graphs on n vertices, respectively. The outer-planar anti-Ramsey number of graph H, denoted by ar(On,H), is defined as max{ar(On,H)|On∈On}. It seems nontrivial to determine the exact values for ar(On,H) because most maximal outer-planar graphs are asymmetry. In this paper, we obtain that ar(On,kK2)≤n+3k−8 for all n≥2k and k≥6, which improves the existing upper bound for ar(On,kK2), and prove that ar(On,kK2)=n+2k−5 for n=2k and k≥5. We also obtain that ar(On,6K2)=n+6 for all n≥29.

Anti-Ramsey numbers for cycles in the generalized Petersen graphs

For H⊆G, the anti-Ramsey number ar(G,H) is the maximum number of colors in an edge-coloring of G such that each subgraph isomorphic to H has at least two edges in the same color. The study of anti-Ramsey number ar(Kn,H) was introduced by Erdős et al. in 1973, and plentiful results were researched for some special graph H. Later, the problem was extended to ar(G,H) when replacing Kn by other graph G such as hypergraph, complete split graph, regular bipartite graph, triangulation and so on. In this paper, we consider a generalized Petersen graph Pn,k as the host graph and determine the exact anti-Ramsey numbers for cycles C5 and C6 in Pn,k, respectively.

Anti-Ramsey number of matchings in r-partite r-uniform hypergraphs

An edge-colored hypergraph is rainbow if all of its edges have different colors. Given two hypergraphs H and G, the anti-Ramsey number ar(G,H) of H in G is the maximum number of colors in a coloring of the edges of G so that there does not exist a rainbow copy of H. Li et al. determined the anti-Ramsey number of k-matchings in complete bipartite graphs. Jin and Zang showed the uniqueness of the extremal coloring. In this paper, as a generalization of these results, we determine the anti-Ramsey number arr(Kn1,…,nr,Mk) of k-matchings in complete r-partite r-uniform hypergraphs and show the uniqueness of the extremal coloring. Also, we show that Kk−1,n2,…,nr is the unique extremal hypergraph for Turán number exr(Kn1,…,nr,Mk) and show that arr(Kn1,…,nr, Mk)=exr(Kn1,…,nr,Mk−1)+1, which gives a multi-partite version result of Özkahya and Young's conjecture.

Improved bounds for anti-Ramsey numbers of matchings in outer-planar graphs

Let On be the set of all maximal outer-planar graphs of order n. Let ar(On,F) denote the maximum positive integer k such that there is a k-edge-coloring of a graph T in the family On which has no rainbow subgraph F. Denote by Mk a matching of size k. In this paper, we prove that ar(On,Mk)≤n+4k−9 for n≥3k−3, which expressively improves the existing upper bound for ar(On,Mk). We also prove that ar(On,M5)=n+4 for all n≥15.

Anti-Ramsey Hypergraph Numbers

The anti-Ramsey number a r n ( H ) of an r-uniform hypergraph is the maximum number of colors that can be used to color the hyperedges of a complete r-uniform hypergraph on n vertices without producing a rainbow copy of H. In this paper, we determine anti-Ramsey numbers for paths of length 2, certain stars and complete hypergraphs, and the complete 3-uniform hypergraph of order 4 with a single hyperedge removed.

Anti-Ramsey number of matchings in a hypergraph

Given an edge-coloring of a hypergraph G, G is said to be rainbow if any two edges of G receive different colors. The anti-Ramsey numberAR(G,H) of a hypergraph H in a hypergraph G is defined to be the maximum integer k such that there exists a k-edge-coloring of G avoiding rainbow copies of H. The anti-Ramsey numbers of matchings have been extensively studied in several graph classes. But there is few results in hypermatchings and up to now, we only know the anti-Ramsey number of matchings, paths and cycles in complete uniform hypergraphs. In this paper, we determine the exact value of the anti-Ramsey number of a k-matching in a complete tripartite 3-uniform hypergraph. Interestingly, our results are parallel to the results in complete bipartite graphs.