The existence of cycles with a given length is classical topic in graph theory with a plethora of open problems. Examples related to the main result of this paper include a conjecture of Burr and Erdős from 1976 asked whether for every integer $m$ and a positive odd integer $k$, there exists $d$ such that every graph with average degree at least $d$ contains a cycle of length $m$ modulo $k$; this conjecture was proven by Bollobás in [Bull. London Math. Soc. 9 (1977), 97-98]( https://doi.org/10.1112/blms/9.1.97). Another example is a problem of Erdős from the 1990s asking whether there exists $A\subseteq\mathbb{N}$ with zero density and constants $n_0$ and $d_0$ such that every graph with at least $n_0$ vertices and the average degree at least $d_0$ contains a cycle with length in the set $A$, which was resolved by Verstraete in [J. Graph Theory 49 (2005), 151-167]( https://doi.org/10.1002/jgt.20072). In 1983, Thomassen conjectured that for all integers $m$ and $k$, every graph with minimum degree $k+1$ contains a cycle of length $2m$ modulo $k$. Note that the parity condition in the first and the third conjectures is necessary because of bipartite graphs. The current paper contributes to this long line of research by proving that for every integer $m$ and a positive odd integer $k$, every sufficiently large $3$-connected cubic graph contains a cycle of length $m$ modulo $k$. The result is the best possible in the sense that the same conclusion is not true for $2$-connected cubic graphs or $3$-connected graphs with minimum degree three.

We prove that, for every natural number k, every sufficiently large 3-connected cubic planar graph has a cycle whose length is in [k,2k+9]. We also show that this bound is close to being optimal by constructing, for every even k≥4, an infinite family of 3-connected cubic planar graphs that contain no cycle whose length is in [k,2k+1].

We construct an infinite family of counterexamples to Thomassen's conjecture that the vertices of every 3-connected, cubic graph on at least 8 vertices can be colored blue and red such that the blue subgraph has maximum degree at most 1 and the red subgraph minimum degree at least 1 and contains no path on 4 vertices.

We introduce the notion of \emph{bounded diameter arboricity}. Specifically, the \emph{diameter-$d$ arboricity} of a graph is the minimum number $k$ such that the edges of the graph can be partitioned into $k$ forests each of whose components has diameter at most $d$. A class of graphs has bounded diameter arboricity $k$ if there exists a natural number $d$ such that every graph in the class has diameter-$d$ arboricity at most $k$. We conjecture that the class of graphs with arboricity at most $k$ has bounded diameter arboricity at most $k+1$. We prove this conjecture for $k\in \{2,3\}$ by proving the stronger assertion that the union of a forest and a star forest can be partitioned into two forests of diameter at most 18. We use these results to characterize the bounded diameter arboricity for planar graphs of girth at least $g$ for all $g\ne 5$. As an application we show that every 6-edge-connected planar (multi)graph contains two edge-disjoint $\frac{18}{19}$-thin spanning trees.

Albertson, Berman, Hutchinson, and Thomassen showed in 1990 that there exist highly connected graphs in which every spanning tree contains vertices of degree 2. Using a result of Alon and Wormald, we show that there exists a natural number d such that every graph of minimum degree at least d contains a spanning tree without adjacent vertices of degree 2. Moreover, we prove that every graph with minimum degree at least 3 has a spanning tree without three consecutive vertices of degree 2.

We prove that every connected graph can be edge-decomposed into a spanning tree, an even graph, and a star forest.

A graph is locally irregular if no two adjacent vertices have the same degree. The irregular chromatic index chi'(irr)(G) of a graph G is the smallest number of locally irregular subgraphs needed to edge-decompose G. Not all graphs have such a decomposition, but Baudon, Bensmail, Przybylo, and Wozniak conjectured that if G can be decomposed into locally irregular subgraphs, then chi'(irr)(G) <= 3. In support of this conjecture, Przybylo showed that chi'(irr)(G) <= 3 holds whenever G has minimum degree at least 10(10).Here we prove that every bipartite graph G which is not an odd length path satisfies chi'(irr)(G) <= 10. This is the first general constant upper bound on the irregular chromatic index of bipartite graphs. Combining this result with Przybylo's result, we show that chi'(irr)(G) <= 328 for every graph G which admits a decomposition into locally irregular subgraphs. Finally, we show that chi'(irr)(G) <= 2 for every 16-edge-connected bipartite graph G. (C) 2016 Elsevier Ltd. All rights reserved.

The Barát–Thomassen conjecture asserts that for every tree T on m edges, there exists a constant kT such that every kT-edge-connected graph with size divisible by m can be edge-decomposed into copies of T. So far this conjecture has only been verified when T is a path or when T has diameter at most 4. Here we prove the full statement of the conjecture.

A dynamic colouring of a graph is a proper colouring in which no neighbourhood of a non-leaf vertex is monochromatic. The dynamic colouring number χ_2(G) of a graph G is the least number of colours needed for a dynamic colouring of G. Montgomery conjectured that χ_2(G) ≤χ(G) + 2 for all regular graphs G, which would significantly improve the best current upper bound χ_2(G) ≤ 2χ(G). In this note, however, we show that this last upper bound is sharp by constructing, for every integer n ≥ 2, a regular graph G with χ(G) = n but χ_2(G) = 2n. In particular, this disproves Montgomery's conjecture.

The Tree Decomposition Conjecture by Barát and Thomassen states that for every tree T there exists a natural number k(T) such that the following holds: If G is a k(T)-edge-connected simple graph with size divisible by the size of T, then G can be edge-decomposed into subgraphs isomorphic to T. So far this conjecture has only been verified for paths, stars, and a family of bistars. We prove a weaker version of the Tree Decomposition Conjecture, where we require the subgraphs in the decomposition to be isomorphic to graphs that can be obtained from T by vertex-identifications. We call such a subgraph a homomorphic copy of T. This implies the Tree Decomposition Conjecture under the additional constraint that the girth of G is greater than the diameter of T. As an application, we verify the Tree Decomposition Conjecture for all trees of diameter at most 4.

An old conjecture by Jünger, Reinelt and Pulleyblank states that every 2-edge-connected planar graph can be decomposed into paths of length 3 and triangles, provided its size is divisible by 3. We prove the conjecture for a class of planar graphs including all 2-edge-connected series-parallel graphs. We also present a 2-edge-connected non-planar graph that can be embedded on the torus and admits no decomposition into paths of length 3 and triangles.