Monochromatic Cycle Partitions in Local Edge Colorings

An edge coloring of a graph is said to be an r‐local coloring if the edges incident to any vertex are colored with at most r colors. Generalizing a result of Bessy and Thomassé, we prove that the vertex set of any 2‐locally colored complete graph may be partitioned into two disjoint monochromatic cycles of different colors. Moreover, for any natural number r, we show that the vertex set of any r‐locally colored complete graph may be partitioned into O(r^(2) log r) disjoint monochromatic cycles. This generalizes a result of Erdős, Gyárfás, and Pyber

Partitioning a graph into monochromatic connected subgraphs

We show that every 2-edge‐colored graph on vertices with minimum degree at least\frac{2n - 5}{3} can be partitioned into two monochromatic connected subgraphs, provided

Minimum degree conditions for monochromatic cycle partitioning

A classical result of Erd\H{o}s, Gy\'arf\'as and Pyber states that any $r$-edge-coloured complete graph has a partition into $O(r^2 \log r)$ monochromatic cycles. Here we determine the minimum degree threshold for this property. More precisely, we show that there exists a constant $c$ such that any $r$-edge-coloured graph on $n$ vertices with minimum degree at least $n/2 + c \cdot r \log n$ has a partition into $O(r^2)$ monochromatic cycles. We also provide constructions showing that the minimum degree condition and the number of cycles are essentially tight.Comment: 22 pages (26 including appendix

Monochromatic cycle partitions in local edge colorings

An edge colouring of a graph is said to be an r-local colouring if the edges incident to any vertex are coloured with at most r colours. Generalising a result of Bessy and Thomassé, we prove that the vertex set of any 2-locally coloured complete graph may be partitioned into two disjoint monochromatic cycles of different colours. Moreover, for any natural number r, we show that the vertex set of any r-locally coloured complete graph may be partitioned into O(r2 log r) disjoint monochromatic cycles. This generalises a result of Erdős, Gyárfás and Pyber.</p

Monochromatic cycle partitions in local edge colorings

An edge colouring of a graph is said to be an r-local colouring if the edges incident to any vertex are coloured with at most r colours. Generalising a result of Bessy and Thomassé, we prove that the vertex set of any 2-locally coloured complete graph may be partitioned into two disjoint monochromatic cycles of different colours. Moreover, for any natural number r, we show that the vertex set of any r-locally coloured complete graph may be partitioned into O(r2 log r) disjoint monochromatic cycles. This generalises a result of Erdős, Gyárfás and Pyber.</p