Due to Veblen's Theorem, if a connected multigraph $X$ has even degrees at each vertex, then it is Eulerian and its edge set has a partition into cycles. In this paper, we show that an Eulerian multigraph has a unique partition into cycles if and only if it belongs to the family $\mathcal{S}$, ``bridgeless cactus multigraphs", elements of which are obtained by replacing every edge of a tree with a cycle of length $\geq 2$. Other characterizing conditions for bridgeless cactus multigraphs and digraphs are provided.
Furthermore, for a digraph $D$, we list conditions equivalent to having a unique Eulerian circuit, thereby generalizing a previous result of Arratia-Bollobás-Sorkin. In particular, we show that digraphs with a unique Eulerian circuit constitute a subfamily of $\mathcal{S}$, namely, ``Christmas cactus digraphs".
We construct new resolvable decompositions of complete multigraphs and complete equipartite multigraphs into cycles of variable lengths (and a perfect matching if the vertex degrees are odd). We develop two techniques: {\em layering}, which allows us to obtain 2-factorizations of complete multigraphs from existing 2-factorizations of complete graphs, and {\em detachment}, which allows us to construct resolvable cycle decompositions of complete equipartite multigraphs from existing resolvable cycle decompositions of complete multigraphs. These techniques are applied to obtain new 2-factorizatio…
We establish necessary and sufficient conditions for the existence of a decomposition of a complete multigraph into edge-disjoint cycles of specified lengths, or into edge-disjoint cycles of specified lengths and a perfect matching.
Bryant, Horsley, Maenhaut and Smith recently gave necessary and sufficient conditions for when the complete multigraph can be decomposed into cycles of specified lengths $m_1,m_2,\ldots,m_τ$. In this paper we characterise exactly when there exists a packing of the complete multigraph with cycles of specified lengths $m_1,m_2,\ldots,m_τ$. While cycle decompositions can give rise to packings by removing cycles from the decomposition, in general it is not known when there exists a packing of the complete multigraph with cycles of various specified lengths.
In 1963, Corrádi and Hajnal proved that for all $k \ge 1$ and $n \ge 3k$, every (simple) graph on n vertices with minimum degree at least 2k contains k disjoint cycles. The same year, Dirac described the 3-connected multigraphs not containing two disjoint cycles and asked the more general question: Which (2k-1)-connected multigraphs do not contain k disjoint cycles? Recently, the authors characterized the simple graphs G with minimum degree $δ(G) \ge 2k-1$ that do not contain k disjoint cycles. We use this result to answer Dirac's question in full.
A $c$-edge-colored multigraph has each edge colored with one of the $c$ available colors and no two parallel edges have the same color. A proper Hamiltonian cycle is a cycle containing all the vertices of the multigraph such that no two adjacent edges have the same color. In this work we establish sufficient conditions for a multigraph to have a proper Hamiltonian cycle, depending on several parameters such as the number of edges and the rainbow degree.
In a graph $G$, let $μ_G(xy)$ denote the number of edges between $x$ and $y$ in $G$. Let $λK_{v,u}$ be the graph $(V\cup U,E)$ with $|V|=v$, $|U|=u$, and \[ μ_G(xy)=\begin{cases} λ&\mbox{if $x\in U$ and $y\in V$ or if $x\in V$ and $y\in U$}\\ 0 &\mbox{otherwise.} \\ \end{cases} \] Let $M$ be a sequence of non-negative integers $m_1,m_2,\ldots,m_n$. An $(M)$-cycle decomposition of a graph $G$ is a partition of the edge set into cycles of lengths $m_1,m_2,\ldots,m_n$. In this paper, we establish necessary and sufficient conditions for the existence of an $(M)$-cycle decomposition of $λK_{v,u}$.
A Hamiltonian decomposition of a regular graph is a partition of its edge set into Hamiltonian cycles. The problem of finding edge-disjoint Hamiltonian cycles in a given regular graph has many applications in combinatorial optimization and operations research. Our motivation for this problem comes from the field of polyhedral combinatorics, as a sufficient condition for vertex nonadjacency in the 1-skeleton of the traveling salesperson polytope can be formulated as the Hamiltonian decomposition problem in a 4-regular multigraph with one forbidden decomposition. In our approach, the algorithm s…
Let $H$ be a graph possibly with loops and $G$ be a multigraph without loops. An $H$-coloring of $G$ is a function $c: E(G) \rightarrow V(H)$. We will say that $G$ is an $H$-colored multigraph, whenever we are taking a fixed $H$-coloring of $G$. The set of all the edges with end vertices $u$ and $v$ will be denoted by $E_{uv}$. We will say that $W=(v_0,e_0^1, \ldots, e_0^{k_0},v_1,e_1^1,\ldots,e_1^{k_1},v_2,\ldots,v_{n-1},e_{n-1}^1,\ldots,e_{n-1}^{k_{n-1}},v_n)$, where for each $i$ in $\{0,\ldots,n-1\}$, $k_i \geq 1$ and $e_i^j \in E_{v_iv_{i+1}}$ for every $j \in \{1,\ldots, k_i \}$, is a dyn…
Using the technique of amalgamation-detachment, we show that the complete equipartite multigraph $λK_{n\times m}$ can be decomposed into cycles of lengths $c_1m,\dots,c_km$ (plus a 1-factor if the degree is odd) whenever there exists a decomposition of $λm K_n$ into cycles of lengths $c_1, \dots,c_k$ (plus a 1-factor if the degree is odd). In addition, we give sufficient conditions for the existence of some other, related cycle decompositions of the complete equipartite multigraph $λK_{n\times m}$.
A mixed multigraph is obtained from an undirected multigraph by orienting a subset of its edges. In this paper, we study a new Hermitian matrix representation of mixed multigraphs, give an introduction to cospectral operations on mixed multigraphs, and characterize switching equivalent mixed multigraphs in terms of fundamental cycle basis. As an application, an upper bound of cospectral classes of mixed multigraphs with the same underlying graph is obtained.
In 1996, Michael Stiebitz proved that if $G$ is a simple graph with $δ(G)\geq s+t+1$ and $s,t\in \mathbb{Z}_{\geq 0}$, then $V(G)$ can be partitioned into two sets $A$ and $B$ such that $δ(G[A])\geq s$ and $δ(G[B])\geq t$. In 2016, Amir Ban proved a similar result for weighted graphs. Let $G$ be a simple graph with at least two vertices, let $w:E(G) \to \mathbb{r}_{>0}$ be a weight function, let $s,t \in \mathbb{R}_{\geq 0}$, and let $W=\max_{e\in E(G)} w(e)$. If $δ(G)\geq s+t+2W$, then $V(G)$ can be partitioned into two sets $A$ and $B$ such that $δ(G[A])\geq s$ and $δ(G[B])\geq t$. This moti…
We characterise the quartic (i.e. 4-regular) multigraphs with the property that every edge lies in a triangle. The main result is that such graphs are either squares of cycles, line multigraphs of cubic multigraphs, or are obtained from these by a number of simple subgraph-replacement operations. A corollary of this is that a simple quartic graph with every edge in a triangle is either the square of a cycle, the line graph of a cubic graph or a graph obtained from the line multigraph of a cubic multigraph by replacing triangles with copies of K_{1,1,3}.
The adaptable choosability of a multigraph $G$, denoted $\mathrm{ch}_a(G)$, is the smallest integer $k$ such that any edge labelling, $τ$, of $G$ and any assignment of lists of size $k$ to the vertices of $G$ permits a list colouring, $σ$, of $G$ such that there is no edge $e = uv$ where $τ(e) = σ(u) = σ(v)$. Here we show that for a multigraph $G$ with maximum degree $Δ$ and no cycles of length 3 or 4, $\mathrm{ch}_a(G) \leq (2\sqrt{2}+o(1))\sqrt{Δ/\lnΔ}$. Under natural restrictions we can show that the same bound holds for the conflict choosability of $G$, which is a closely related parameter…
This paper has been withdrawn by the author. Peterson and Woodall previously proved that the list-edge-colouring conjecture holds for graphs without odd cycles of length 5 or longer. D. Peterson and D. R. Woodall, Edge-choosability in line-perfect multigraphs, Discrete Mathematics 202 (1999), 191-199. D. Peterson and D. R. Woodall, Erratum to "Edge-choosability in line-perfect multigraphs", Discrete Mathematics 260 (2003), 323-326.
In this article, we introduce the notion of the double competition multigraph of a digraph. We give characterizations of the double competition multigraphs of arbitrary digraphs, loopless digraphs, reflexive digraphs, and acyclic digraphs in terms of edge clique partitions of the multigraphs.
The main topic considered is maximizing the number of cycles in a graph with given number of edges. In 2009, Király conjectured that there is constant $c$ such that any graph with $m$ edges has at most $(1.4)^m$ cycles. In this paper, it is shown that for sufficiently large $m$, a graph with $m$ edges has at most $(1.443)^m$ cycles. For sufficiently large $m$, examples of a graph with $m$ edges and $(1.37)^m$ cycles are presented. For a graph with given number of vertices and edges an upper bound on the maximal number of cycles is given. Also, exponentially tight bounds are proved for the maxi…
A locally irregular multigraph is a multigraph whose adjacent vertices have distinct degrees. The locally irregular edge coloring is an edge coloring of a multigraph $G$ such that every color induces a locally irregular submultigraph of $G$. We say that a multigraph $G$ is locally irregular colorable if it admits a locally irregular edge coloring and we denote by ${\rm lir}(G)$ the locally irregular chromatic index of $G$, which is the smallest number of colors required in a locally irregular edge coloring of a locally irregular colorable multigraph $G$. We conjecture that for every connected …
An edge colouring of a multigraph can be thought of as a partition of the edges into matchings (a matching meets each vertex at most once). Analogously, an edge cover colouring is a partition of the edges into edge covers (an edge cover meets each vertex at least once). We aim to determine a tight lower bound on the maximum number of parts in an edge cover colouring as a function of the minimum degree delta, which would be an analogue of Shannon's theorem from 1949 on edge-colouring multigraphs. We are able to give a lower bound that is tight except when delta=9 or delta is odd and > 12; in th…
We consider a Hamiltonian decomposition problem of partitioning a regular graph into edge-disjoint Hamiltonian cycles. A sufficient condition for vertex adjacency in the 1-skeleton of the traveling salesperson polytope can be formulated as the Hamiltonian decomposition problem in a 4-regular multigraph. We introduce a heuristic general variable neighborhood search algorithm for this problem based on finding a vertex-disjoint cycle cover of the multigraph through reduction to perfect matching and several cycle merging operations. The algorithm has a one-sided error: the answer "not adjacent" is…
The star chromatic index of a multigraph $G$, denoted $χ'_{s}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. A multigraph $G$ is star $k$-edge-colorable if $χ'_{s}(G)\le k$. Dvořák, Mohar and Šámal [Star chromatic index, J Graph Theory 72 (2013), 313--326] proved that every subcubic multigraph is star $7$-edge-colorable, and conjectured that every subcubic multigraph should be star $6$-edge-colorable. Kerdjoudj, Kostochka and Raspaud considered the list version of this problem for simple graphs and proved …
The star chromatic index of a multigraph $G$, denoted $χ'_{s}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. A multigraph $G$ is star $k$-edge-colorable if $χ'_{s}(G)\le k$. Dvořák, Mohar and Šámal [Star chromatic index, J. Graph Theory 72 (2013), 313--326] proved that every subcubic multigraph is star $7$-edge-colorable. They conjectured in the same paper that every subcubic multigraph should be star $6$-edge-colorable. In this paper, we first prove that it is NP-complete to determine whether $χ'_s(G)\le3…
In this paper, we prove that, given a clique-width $k$-expression of an $n$-vertex graph, \textsc{Hamiltonian Cycle} can be solved in time $n^{\mathcal{O}(k)}$. This improves the naive algorithm that runs in time $n^{\mathcal{O}(k^2)}$ by Espelage et al. (WG 2001), and it also matches with the lower bound result by Fomin et al. that, unless the Exponential Time Hypothesis fails, there is no algorithm running in time $n^{o(k)}$ (SIAM. J. Computing 2014). We present a technique of representative sets using two-edge colored multigraphs on $k$ vertices. The essential idea is that, for a two-edge c…
Let $k$, $λ$ and $μ$ be positive integers. A decomposition of a multigraph $ λG$ into edge-disjoint subgraphs $G_1, \ldots , G_k$ is said to be \emph{enclosed} by a decomposition of a multigraph $μH$ into edge-disjoint subgraphs $H_1, \ldots , H_k$ if $μ> λ$ and $G_i$ is a subgraph of $H_i$, $1 \leq i \leq k$. In this paper we initiate the study of when a decomposition can be enclosed by a decomposition that consists of spanning subgraphs. A decomposition of a graph is a 2-factorization if each subgraph is 2-regular and is Hamiltonian if each subgraph is a Hamiltonian cycle. Let $n$ and $m$ be…
The arboricity of a graph G is the minimum number of colours needed to colour the edges of G so that every cycle gets at least two colours. Given a positive integer p, we define the generalized p-arboricity Arb_p(G) of a graph G as the minimum number of colours needed to colour the edges of a multigraph G in such a way that every cycle C gets at least min(|C|; p + 1) colours. In the particular case where G has girth at least p + 1, Arb_p(G) is the minimum size of a partition of the edge set of G such that the union of any p parts induce a forest. If we require further that the edge colouring b…
Let G = (V, E) be a multigraph without loops and for any x {\in}V let E(x) be the set of edges of G incident to x. A homogeneous edge-coloring of G is an assignment of an integer m >= 2 and a coloring c:E {\to} S of the edges of Gsuchthat|S| = mandforanyx{\in}V,if|E(x)| = mqx+rx with0 <= rx <m, there exists a partition of E(x) in rx color classes of cardinality qx + 1 and other m-rx color classes of cardinality qx. The homogeneous chromatic index \c{hi}(G) is the least m for which there exists such a coloring. We determine \c{hi}(G) in the case that G is a complete multigraph, a tree or a comp…
A decomposition of a multigraph $G$ is a partition of its edges into subgraphs $G(1), \ldots , G(k)$. It is called an $r$-factorization if every $G(i)$ is $r$-regular and spanning. If $G$ is a subgraph of $H$, a decomposition of $G$ is said to be enclosed in a decomposition of $H$ if, for every $1 \leq i \leq k$, $G(i)$ is a subgraph of $H(i)$. Feghali and Johnson gave necessary and sufficient conditions for a given decomposition of $λK_n$ to be enclosed in some $2$-edge-connected $r$-factorization of $μK_{m}$ for some range of values for the parameters $n$, $m$, $λ$, $μ$, $r$: $r=2$, $μ>λ$ an…
We consider a Hamiltonian decomposition problem of partitioning a regular graph into edge-disjoint Hamiltonian cycles. It is known that verifying vertex non-adjacency in the 1-skeleton of the symmetric and asymmetric traveling salesperson polytopes is NP-complete. On the other hand, a sufficient condition for two vertices to be non-adjacent can be formulated as a combinatorial problem of finding a second Hamiltonian decomposition of a 4-regular multigraph. We present two backtracking algorithms for constructing a second Hamiltonian decomposition and verifying vertex non-adjacency: an algorithm…
A multigraph G is triangle decomposable if its edge set can be partitioned into subsets, each of which induces a triangle of G, and rationally triangle decomposable if its triangles can be assigned rational weights such that for each edge e of G, the sum of the weights of the triangles that contain e equals 1. We present a necessary and sufficient condition for a planar multigraph to be triangle decomposable. We also show that if a simple planar graph is rationally triangle decomposable, then it has such a decomposition using only weights 0,1 and 1/2. This result provides a characterization of…
Orientations of $10$-Edge-Connected Planar Multigraphs and Applications
William & Mary, Department of Mathematics（威廉与玛丽学院）；School of Mathematical Sciences and LPMC, Nankai University（南开大学数学科学学院）；Henan University of Engineering（河南工程学院）
A multigraph is a nonsimple graph which is permitted to have multiple edges, that is, edges that have the same end nodes. We introduce the concept of spanning simplicial complexes $Δ_s(\mathcal{G})$ of multigraphs $\mathcal{G}$, which provides a generalization of spanning simplicial complexes of associated simple graphs. We give first the characterization of all spanning trees of a uni-cyclic multigraph $\mathcal{U}_{n,m}^r$ with $n$ edges including $r$ multiple edges within and outside the cycle of length $m$. Then, we determine the facet ideal $I_\mathcal{F}(Δ_s(\mathcal{U}_{n,m}^r))$ of spa…
In this paper, we consider the decomposition of multigraphs under minimum degree constraints and give a unified generalization of several results by various researchers. Let $G$ be a multigraph in which no quadrilaterals share edges with triangles and other quadrilaterals and let $μ_G(v)=\max\{μ_G(u,v):u\in V(G)\setminus\{v\}\}$, where $μ_G(u,v)$ is the number of edges joining $u$ and $v$ in $G$. We show that for any two functions $a,b:V(G)\rightarrow\mathbb{N}\setminus\{0,1\}$, if $d_G(v)\ge a(v)+b(v)+2μ_G(v)-3$ for each $v\in V(G)$, then there is a partition $(X,Y)$ of $V(G)$ such that $d_X(…
A mixed multigraph is a multigraph which may contain both undirected and directed edges. An orientation of a mixed multigraph $G$ is an assignment of exactly one direction to each undirected edge of $G$. A mixed multigraph $G$ can be oriented to a strongly connected digraph if and only if $G$ is bridgeless and strongly connected [Boesch and Tindell, Am. Math. Mon., 1980]. For each $r \in \mathbb{N}$, let $f(r)$ denote the smallest number such that any strongly connected bridgeless mixed multigraph with radius $r$ can be oriented to a digraph of radius at most $f(r)$. We improve the current bes…
A graph $G$ is $(I,F)$-partitionable if its vertex set can be partitioned into two parts such that one part is an independent set, and the other induces a forest. In this paper, we prove that every planar graph without cycles of length $4, 6, 9$ is $(I,F)$-partitionable.
The star chromatic index of a multigraph $G$, denoted $χ'_{st}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bicolored. We survey the results of determining the star chromatic index, present the interesting proofs and techniques, and collect many open problems and conjectures.
The notion of noncrossing linked partition arose from the study of certain transforms in free probability theory. It is known that the number of noncrossing linked partitions of [n+1] is equal to the n-th large Schroder number $r_n$, which counts the number of Schroder paths. In this paper we give a bijective proof of this result. Then we introduce the structures of linked partitions and linked cycles. We present various combinatorial properties of noncrossing linked partitions, linked partitions, and linked cycles, and connect them to other combinatorial structures and results, including incr…
A classical result of Erdős, Gyárfás 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.
We show that for any colouring of the edges of the complete bipartite graph $K_{n,n}$ with 3 colours there are 5 disjoint monochromatic cycles which together cover all but $o(n)$ of the vertices. In the same situation, 18 disjoint monochromatic cycles together cover all vertices.
Nash-Williams proved that for an undirected graph $ G $ the set $ E(G) $ can be partitioned into cycles if and only if every cut has either even or infinite number of edges. Later C. Thomassen gave a simpler proof for this and conjectured the following directed analogue of the theorem: the edge set of a digraph can be partitioned into directed cycles if and only if for each subset of the vertices the cardinality of the ingoing and the outgoing edges are equal. The aim of the paper is to prove this conjecture.
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 $O(r^2 \log r)$ disjoint monochromatic cycles. This generalises a result of Erdős, Gyárfás and Pyber.
Balogh, Barát, Gerbner, Gyárfás, and Sárközy proposed the following conjecture. Let $G$ be a graph on $n$ vertices with minimum degree at least $3n/4$. Then for every $2$-edge-colouring of $G$, the vertex set $V(G)$ may be partitioned into two vertex-disjoint cycles, one of each colour. We prove that this conjecture holds for $n$ large enough, improving approximate results by the aforementioned authors and by DeBiasio and Nelsen.
Magnant and Martin conjectured that the vertex set of any $d$-regular graph $G$ on $n$ vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = Ω(n)$, improving a result of Han, who showed that in this range almost all vertices of $G$ can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact, our proof gives a partition of $V(G)$ into cycles. We also show that, if $d = Ω(n)$ and $G$ is bipartite, then $V(G)$ can be partitioned into $n / (2d)$ paths (this bound in tight for b…
Confirming a conjecture of Gyárfás, we prove that, for all natural numbers $k$ and $r$, the vertices of every $r$-edge-coloured complete $k$-uniform hypergraph can be partitioned into a bounded number (independent of the size of the hypergraph) of monochromatic tight cycles. We further prove that, for for all natural numbers $p$ and $r$, the vertices of every $r$-edge-coloured complete graph can be partitioned into a bounded number of $p$-th powers of cycles, settling a problem of Elekes, Soukup, Soukup and Szentmiklóssy. In fact we prove a common generalisation of both theorems which further …
We study universal cycles of the set ${\cal P}(n,k)$ of $k$-partitions of the set $[n]:=\{1,2,\ldots,n\}$ and prove that the transition digraph associated with ${\cal P}(n,k)$ is Eulerian. But this does not imply that universal cycles (or ucycles) exist, since vertices represent equivalence classes of partitions! We use this result to prove, however, that ucycles of ${\cal P}(n,k)$ exist for all $n \geq 3$ when $k=2$. We reprove that they exist for odd $n$ when $k = n-1$ and that they do not exist for even $n$ when $k = n-1$. An infinite family of $(n,k)$ for which ucycles do not exist is show…
Erdős, Gyárfás and Pyber showed that every $r$-edge-coloured complete graph $K_n$ can be covered by $25 r^2 \log r$ vertex-disjoint monochromatic cycles (independent of $n$). Here, we extend their result to the setting of binomial random graphs. That is, we show that if $p = p(n) = Ω(n^{-1/(2r)})$, then with high probability any $r$-edge-coloured $G(n,p)$ can be covered by at most $1000 r^4 \log r $ vertex-disjoint monochromatic cycles. This answers a question of Korándi, Mousset, Nenadov, Škorić and Sudakov.
Let $D=(V,A)$ be a directed graph of order $n\geq 6$. Let $W$ be a subset of $V$ with $|W|\geq 6$. Suppose that every vertex of $W$ has degree at least $(3n-3)/2$ in $D$. Then for any integer partition $|W|=n_1+n_2$ with $n_1\geq 3$ and $n_2\geq 3$, $D$ contains two disjoint directed cycles $C_1$ and $C_2$ such that $|V(C_1)\cap W|=n_1$ and $|V(C_2)\cap W|=n_2$. We conjecture that for any integer partition $|W|=n_1+n_2+\cdots +n_k$ with $k\geq 3$ and $n_i\geq 3(1\leq i\leq k)$, $D$ contains $k$ disjoint directed cycles $C_1,C_2,\ldots , C_k$ such that $|V(C_i)\cap W|=n_i$ for all $1\leq i\leq …
In this paper we investigate results of the form "every graph $G$ has a cycle $C$ such that the induced subgraph of $G$ on $V(G)\setminus V(C)$ has small maximum degree." Such results haven't been studied before, but are motivated by the Bessy and Thomassé Theorem which states that the vertices of any graph $G$ can be covered by a cycle $C_1$ in $G$ and disjoint cycle $C_2$ in the complement of $G$. There are two main theorems in this paper. The first is that every graph has a cycle with $Δ(G[V(G)\setminus V(C)])\leq \frac12(|V(G)\setminus V(C)|-1)$. The bound on the maximum degree $Δ(G[V(G)\s…
We present a family of partitions of $W_\mathcal{G}$, the set of walks on a directed graph $\mathcal{G}$. Each partition in this family is identified by an integer sequence $K$, which specifies a collection of cycles on $\mathcal{G}$ with a certain well-defined structure. We term such cycles resummable, and a walk that does not traverse any such cycles $K$-irreducible. For a given value of $K$, the corresponding partition of $W_\mathcal{G}$ consists of a collection of cells that each contain a single $K$-irreducible walk $i$ plus all walks that can be formed from $i$ by attaching one or more r…
We present results on partitioning the vertices of $2$-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of Sárközy: the vertex set of every $2$-edge-colored graph can be partitioned into at most $2α(G)$ monochromatic cycles, where $α(G)$ denotes the independence number of $G$. Another direction, emerged recently from a conjecture of Schelp, is to consider colorings of graphs with given minimum degree. We prove that apart from $o(|V(G)|)$ vertices, the vertex set of any $2$-edge-colored graph $G$ with minimum degree at least $(1…
In 1982 Thomassen asked whether there exists an integer f(k,t) such that every strongly f(k,t)-connected tournament T admits a partition of its vertex set into t vertex classes V_1,...,V_t such that for all i the subtournament T[V_i] induced on T by V_i is strongly k-connected. Our main result implies an affirmative answer to this question. In particular we show that f(k,t) = O(k^7 t^4) suffices. As another application of our main result we give an affirmative answer to a question of Song as to whether, for any integer t, there exists an integer h(t) such that every strongly h(t)-connected tou…
