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### Rainbow odd cycles [article]

Ron Aharoni, Joseph Briggs, Ron Holzman, Zilin Jiang
2021 arXiv   pre-print
As part of the proof, we characterize those families of n odd cycles in K_n+1 that do not have any rainbow odd cycle.  ...  We prove that every family of (not necessarily distinct) odd cycles O_1, ..., O_2⌈ n/2 ⌉-1 in the complete graph K_n on n vertices has a rainbow odd cycle (that is, a set of edges from distinct O_i's,  ...  odd cycle by a K-rainbow even path in Q.  ...

### Generalized rainbow Turán numbers of odd cycles [article]

József Balogh, Michelle Delcourt, Emily Heath, Lina Li
2021 arXiv   pre-print
This matches the known lower bound for k even and is conjectured to be tight for k odd.  ...  Given graphs F and H, the generalized rainbow Turán number ex(n,F,rainbow-H) is the maximum number of copies of F in an n-vertex graph with a proper edge-coloring that contains no rainbow copy of H.  ...  If k ≥ 2 is odd then ex(n, C 3 , rainbow-C 2k ) = Ω(n 1+1/k ), and if k is even then ex(n, C 3 , rainbow-C 2k+1 ) = Ω(n 1+1/k ).  ...

### On Odd Rainbow Cycles in Edge-Colored Graphs [article]

Andrzej Czygrinow, Theodore Molla, Brendan Nagle, Roy Oursler
2019 arXiv   pre-print
We prove that the same condition ensures a rainbow ℓ-cycle C_ℓ whenever n > 432 ℓ. This result is sharp for all odd integers ℓ≥ 3, and extends earlier work of the authors when ℓ is even.  ...  Li proved that if every vertex v ∈ V is incident to at least (n+1)/2 distinctly colored edges, then G admits a rainbow triangle.  ...  If δ c (G) ≥ (n + 1)/2, then (G, c) admits a rainbow 3-cycle C 3 . A rainbow K ⌊n/2⌋,⌈n/2⌉ establishes that Theorem 1.1 is best possible.  ...

### On lengths of rainbow cycles [article]

Boris Alexeev
2006 arXiv   pre-print
We settle a question posed by Ball, Pultr, and Vojtěchovský by showing that if such a coloring does not contain a rainbow cycle of length n, where n is odd, then it also does not contain a rainbow cycle  ...  We prove several results regarding edge-colored complete graphs and rainbow cycles, cycles with no color appearing on more than one edge.  ...  First, we must show that there exist rainbow cycles of all odd lengths. But this is easy! Consider the cycle (1, 2, 3, . . . , k) for k odd.  ...

### On Lengths of Rainbow Cycles

Boris Alexeev
2006 Electronic Journal of Combinatorics
We settle a question posed by Ball, Pultr and Vojtěchovský by showing that if such a coloring does not contain a rainbow cycle of length \$n\$, where \$n\$ is odd, then it also does not contain a rainbow cycle  ...  We prove several results regarding edge-colored complete graphs and rainbow cycles, cycles with no color appearing on more than one edge.  ...  m-cycle but no rainbow n-cycle.  ...

### Rainbow Cycles in Flip Graphs

Stefan Felsner, Linda Kleist, Torsten Mütze, Leon Sering, Marc Herbstritt
2018 International Symposium on Computational Geometry
An r-rainbow cycle in this graph is a cycle in which every inner edge of the triangulation appears exactly r times. This notion of a rainbow cycle extends in a natural way to other flip graphs.  ...  In each of the five settings, we prove the existence and non-existence of rainbow cycles for different values of r, n and k.  ...  Acknowledgements We thank Manfred Scheucher for his quick assistance in running computer experiments that helped us to find rainbow cycles in small flip graphs.  ...

### A Polynomial Time Algorithm for Computing the Strong Rainbow Connection Numbers of Odd Cacti [article]

Logan A. Smith, David T. Mildebrath, Illya V. Hicks
2019 arXiv   pre-print
We consider the problem of computing the strong rainbow connection number src(G) for cactus graphs G in which all cycles have odd length.  ...  We present a formula to calculate src(G) for such odd cacti which can be evaluated in linear time, as well as an algorithm for computing the corresponding optimal strong rainbow edge coloring, with polynomial  ...  Additionally, we note that while odd length cycles are odd cacti, a graph can be identified as an odd length cycle in worst case linear time complexity and the strong rainbow connection numbers of cycles  ...

### Rainbow regular order of graphs

Zdzislaw Skupien, Andrzej Zak
2008 The Australasian Journal of Combinatorics
Hence color classes are maximum matchings rotationally/cyclically generated if t is even/odd. A rainbow subgraph of GK t has all edges with distinct colors.  ...  We have solved this problem for cycles, wheels and complete bipartite graphs in case G is a complete graph.  ...  For instance, odd cycles are sequential [11] and even cycles are elegant [5, 17] but not harmonious [12] . Nevertheless, the following result requires a proof.  ...

### Generalisation of the rainbow neighbourhood number and 𝑘-jump colouring of a graph

Johan Kok, Sudev Naduvath, Eunice Gogo Mphako-Banda
2020 Annales Mathematicae et Informaticae
In this paper, the notions of rainbow neighbourhood and rainbow neighbourhood number of a graph are generalised and further to these generalisations, the notion of a proper -jump colouring of a graph is  ...  If the closed -neighbourhood [ ] contains at least one of each colour of the chromatic colour set, we say that yields a -rainbow neighbourhood.  ...  Consider a graph which contains an odd cycle, , ≤ . Here are two sub-cases to be considered. (a) Assume that has odd cycle and the arbitrary vertex 1 / ∈ ( ).  ...

### Rainbow cycles in flip graphs [article]

Stefan Felsner, Linda Kleist, Torsten Mütze, Leon Sering
2018 arXiv   pre-print
An r-rainbow cycle in this graph is a cycle in which every inner edge of the triangulation appears exactly r times. This notion of a rainbow cycle extends in a natural way to other flip graphs.  ...  In each of the five settings, we prove the existence and non-existence of rainbow cycles for different values of r, n and k.  ...  Acknowledgements We thank Manfred Scheucher for his quick assistance in running computer experiments that helped us to find rainbow cycles in small flip graphs.  ...

### Connected components of meanders: I. bi-rainbow meanders

Anna Karnauhova, Stefan Liebscher
2017 Discrete and Continuous Dynamical Systems. Series A
We address this question in the special case of bi-rainbow meanders, which are given as nonbranched families (rainbows) of nested arcs.  ...  Easily obtainable results for small bi-rainbow meanders containing at most four families in total (lower and upper rainbow families) suggest an expression of the number of curves by the greatest common  ...  Then a product π / of α disjoint transpositions represents a disjoint arc collection if, and only if, the permutation π / σ has exactly α + 1 (disjoint) cycles. π / | {odd} = {even}, π / | {even} = {odd  ...

### RAINBOW CONNECTION NUMBER OF FLOWER SNARK GRAPH

K. Srinivasa Rao, U.V.C. Kumar, A. Mekala
2020 International Journal of Applied Mathematics
Let G = J n , for odd n ≥ 5. Then G is rainbow critical, i.e. rc(G − e) = 2n for n ≥ 5 and odd. Proof. We consider vertex set V (G) and edge set E(G) as defined in Theorem 4.  ...  In the next section, we determine rainbow connection number rc(G) for the flower snark graph J n for n ≥ 5 and odd. Main results Theorem 4. Let G = J n . Then for odd n ≥ 5, rc(G) = ⌈ n 2 ⌉ + 3.  ...

### STRONG RAINBOW EDGE-COLOURING OF VARIANTS OF CUBIC HALIN GRAPHS

I.A. Arputhamary, M.H. Mercy
2017 International Journal of Pure and Applied Mathematics
The strong rainbow connection number of G, denoted by src(G) is the minimum number of colours that makes G strongly rainbow connected.  ...  In this paper we explore the strong rainbow connection number of variants of cubic Halin graphs.  ...  Subcase 2.2: The cycle C R ⌈ n 2 ⌉+3 is odd and the cycle C L ⌈ n 2 ⌉+2 is even.  ...

### Rainbow vertex-connection number of 2-connected graphs [article]

Xueliang Li, Sujuan Liu
2011 arXiv   pre-print
In this paper we first determine the rainbow vertex-connection number of cycle C_n of order n≥ 3, and then, based on it, prove that for any 2-connected graph G, rvc(G)≤ rvc(C_n), giving a tight upper bound  ...  for the rainbow vertex-connection.  ...  In the following, we assume that G is not an odd cycle.  ...

### RAINBOW CONNECTION IN MODIFIED BRICK PRODUCT GRAPHS

K. Srinivasa Rao, R. Murali
2017 Far East Journal of Mathematical Sciences (FJMS)
In this paper we determine rc(G) of brick product graphs associated with even cycles. We also discuss the critical property of these graphs with respect to rainbow coloring.  ...  A path in G is called a rainbow path if no two edges of it are colored the same. G is rainbow connected if G contains a rainbow u − v path for every two vertices u and v in it.  ...  Conclusion In this paper, we have determined the rainbow connection number of brick product graphs C(2n, m, r) associated with even cycles.  ...
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