Connect Four and Graph Decomposition

The existence of planar $4$-connected essentially $6$-edge-connected graphs with no claw-decompositions

In 2006 Bar{á}t and Thomassen conjectured that every planar $4$-edge-connected $4$-regular simple graph of size divisible by three admits a claw-decomposition. Later, Lai (2007) disproved this conjecture by a family of planar graphs with edge-connectivity $4$ which the smallest one contains $24$ vertices. In this note, we first give a smaller counterexample having only $18$ vertices and next construct a family of planar $4$-connected essentially $6$-edge-connected $4$-regular simple graphs of size divisible by three with no claw-decompositions. This result provides the sharpness for two known …

Decompositions of highly connected graphs into paths of length five

We study the Decomposition Conjecture posed by Barát and Thomassen (2006), which states that for every tree $T$ there exists a natural number $k_T$ such that, if $G$ is a $k_T$-edge-connected graph and $|E(T)|$ divides $|E(G)|$, then $G$ admits a decomposition into copies of $T$. In a series of papers, Thomassen verified this conjecture for stars, some bistars, paths of length $3$, and paths whose length is a power of $2$. We verify the Decomposition Conjecture for paths of length $5$.

Decomposition of cubic graphs with cyclic connectivity 5

Let $G$ be a cyclically $5$-connected cubic graph with a $5$-edge-cut separating $G$ into two cyclic components $G_1$ and $G_2$. We prove that each component $G_i$ can be completed to a cyclically $5$-connected cubic graph by adding three vertices, unless $G_i$ is a cycle of length five. Our work extends similar results by Andersen et al. for cyclic connectivity $4$ from 1988.

Uniformly resolvable decompositions of K_v into paths on two, three and four vertices

In this paper we consider uniformly resolvable decompositions of the complete graph K_v into subgraphs such that each resolution class contains only blocks isomorphic to the same graph. We completely determine the spectrum for the case in which all the resolution classes consist of either P_2, P_3 and P_4.

On the decomposition of connected graphs into their biconnected components

We give a recursion formula to generate all equivalence classes of biconnected graphs with coefficients given by the inverses of the orders of their groups of automorphisms. We give a linear map to produce a connected graph with say, u, biconnected components from one with u-1 biconnected components. We use such map to extend the aforesaid result to connected or 2-edge connected graphs. The underlying algorithms are amenable to computer implementation.

关于强连通双向图的Ear分解

On Ear Decompositions of Strongly Connected Bidirected Graphs

双向图（Edmonds、Johnson和Tutte、Goldberg、Karzanov等人在反对称图的等价术语中研究过）被证明是描述流和匹配问题的有用的统一语言。本文将耳分解的概念推广到强连通双向图类。特别地，我们的结果暗示了关于Lovász和Plummer的匹配覆盖图的双耳定理。除了反对称图上的标准Barrier定理外，本文给出的证明都是完备的。

The Four-Vertex Theorem, The Evolute, and The Decomposition of Polygons

The Four-Vertex Theorem has been of interest ever since a discrete version appeared in 1813 due to Cauchy. Up until now, there have been many different versions of this theorem, both for discrete cases and smooth cases. In 2004, an approach relating the discrete Four-Vertex Theorem to the evolute was published, and here we will give an overview of this paper. We then will define the notion of the decomposition of polygons, and derive some new results about how this notion affects various types of extremality. We will see that from our fresh results we can easily derive discrete Four-Vertex The…

Minimum Cycle Decomposition: A Constructive Characterization for Graphs of Treewidth Two with Node Degrees Two and Four

Substantial efforts have been made to compute or estimate the minimum number $c(G)$ of cycles needed to partition the edges of an Eulerian graph. We give an equivalent characterization of Eulerian graphs of treewidth $2$ and with maximum degree $4$. This characterization enables us to present a linear time algorithm for the computation of $c(G)$ for all $G$ in this class.

Edge-decomposition of graphs into copies of a tree with four edges

We study edge-decompositions of highly connected graphs into copies of a given tree. In particular we attack the following conjecture by Barát and Thomassen: for each tree $T$, there exists a natural number $k_T$ such that if $G$ is a $k_T$-edge-connected graph, and $|E(T)|$ divides $|E(G)|$, then $E(G)$ has a decomposition into copies of $T$. As one of our main results it is sufficient to prove the conjecture for bipartite graphs. Let $Y$ be the unique tree with degree sequence $(1,1,1,2,3)$. We prove that if $G$ is a 191-edge-connected graph of size divisible by 4, then $G$ has a $Y$-decompo…

On edge-decomposition of cubic graphs into copies of the double-star with four edges

A tree containing exactly two non-pendant vertices is called a double-star. Let $k_1$ and $k_2$ be two positive integers. The double-star with degree sequence $(k_1+1, k_2+1, 1, \ldots, 1)$ is denoted by $S_{k_1, k_2}$. If $G$ is a cubic graph and has an $S$-decomposition, for a double-star $S$, then $S$ is isomorphic to $S_{1,1}$, $S_{1,2}$ or $S_{2,2}$. It is known that a cubic graph has an $S_{1,1}$-decomposition if and only if it contains a perfect matching. In this paper we study the $S_{1,2}$-decomposition of cubic graphs. First, we present some necessary conditions for the existence of …

耳分解的推广与高度连通无星图中的k-树

A generalization of an ear decomposition and k-trees in highly connected star-free graphs

本文提出$j$-蜘蛛分解，改进高度连通无星图中$k$-树存在的充分条件，证明$j$-连通$K_{1,j(k-2)+2}$-free图有$k$-树，优于Jackson-Wormald结果。

将边连通度为 $O(m)$ 的图边分解为固定大小为 $m$ 的树的同构副本

Edge-decompositions of $O(m)$-edge-connected graphs into isomorphic copies of a fixed tree of size $m$

本文证明在 $O(m)$ 边连通且满足一定最小度或围长条件下，大小为 $m$ 的倍数的简单图可边分解为任意给定大小为 $m$ 的树的同构副本，从而将此前边连通度的阶乘上界改进为线性量级。

On Hamilton Decompositions of Infinite Circulant Graphs

The natural infinite analogue of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph). Although it is known that every connected $2k$-valent infinite circulant graph has a two-way-infinite Hamilton path, there exist many such graphs that do not have a decomposition into $k$ edge-disjoint two-way-infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every $2k$-valent connected circulant graph has a decomposition into $k$ edge-disjoint Hamilton cycles. We settle the problem of decomposing $2k$-valent infinite cir…

A Note on Induced Path Decomposition of Graphs

Let $G$ be a graph of order $n$. The path decomposition of $G$ is a set of disjoint paths, say $\mathcal{P}$, which cover all vertices of $G$. If all paths are induced paths in $G$, then we say $\mathcal{P}$ is an induced path decomposition of $G$. Moreover, if every path is of order at least 2, then we say $G$ has an IPD. In this paper, we prove that every connected $r$-regular graph which is not complete graph of odd order admits an IPD. Also we show that every connected bipartite cubic graph of order $n$ admits an IPD of size at most $\frac{n}{3}$. We classify all connected claw-free graphs…

Decomposing highly edge-connected graphs into homomorphic copies of a fixed tree

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…

A Graph Decomposition motivated by the Geometry of Randomized Rounding

We introduce a graph decomposition which exists for all simple, connected graphs $G=(V,E)$. The decomposition $V = A \cup B \cup C$ is such that each vertex in $A$ has more neighbors in $B$ than in $A$ and vice versa. $C$ is `balanced': each $v \in C$ has the same number of neighbours in $A$ and $B$. These decompositions arise naturally from the behavior of an associated dynamical system (`Randomized Rounding') on $(\mathbb{S}^1)^{|V|}$. Connections to judicious partitions and the \textsc{MaxCut} problem (in particular the Burer-Monteiro-Zhang heuristic) are being discussed.

Gallai's path decomposition in planar graphs

In 1968, Gallai conjectured that the edges of any connected graph with $n$ vertices can be partitioned into $\lceil \frac{n}{2} \rceil$ paths. We show that this conjecture is true for every planar graph. More precisely, we show that every connected planar graph except $K_3$ and $K_5^-$ ($K_5$ minus one edge) can be decomposed into $\lfloor \frac{n}{2} \rfloor$ paths.

Reductions for the 3-Decomposition Conjecture

The 3-decomposition conjecture is wide open. It asserts that every finite connected cubic graph can be decomposed into a spanning tree, a disjoint union of cycles, and a matching. We show that every such decomposition is derived from a homeomorphically irreducible spanning tree (HIST). This allows us to propose a novel reformulation of the 3-decomposition conjecture: the HIST-extension conjecture. We also prove that the following graphs are reducible configurations with respect to the 3-decomposition conjecture: the triangle, the K_{2,3}, the Petersen graph with one vertex removed, the claw-sq…

Bidirected Graphs I: Signed General Kotzig-Lovász Decomposition

This paper is the first from a series of papers that establish a common analogue of the strong component and basilica decompositions for bidirected graphs. A bidirected graph is a graph in which a sign $+$ or $-$ is assigned to each end of each edge, and therefore is a common generalization of digraphs and signed graphs. Unlike digraphs, the reachabilities between vertices by directed trails and paths are not equal in general bidirected graphs. In this paper, we set up an analogue of the strong connectivity theory for bidirected graphs regarding directed trails, motivated by factor theory. We …