Properties of Catlin's reduced graphs and supereulerian graphs

A graph $G$ is called collapsible if for every even subset $R\subseteq V(G)$, there is a spanning connected subgraph $H$ of $G$ such that $R$ is the set of vertices of odd degree in $H$. A graph is the reduction of $G$ if it is obtained from $G$ by contracting all the nontrivial collapsible subgraphs. A graph is reduced if it has no nontrivial collapsible subgraphs. In this paper, we first prove a few results on the properties of reduced graphs. As an application, for 3-edge-connected graphs $G$ of order $n$ with $d(u)+d(v)\ge 2(n/p-1)$ for any $uv\in E(G)$ where $p>0$ are given, we show how such graphs change if they have no spanning Eulerian subgraphs when $p$ is increased from $p=1$ to 10 then to $15$

Optimal path and cycle decompositions of dense quasirandom graphs

Motivated by longstanding conjectures regarding decompositions of graphs into paths and cycles, we prove the following optimal decomposition results for random graphs. Let $0<p<1$ be constant and let $G\sim G_{n,p}$. Let $odd(G)$ be the number of odd degree vertices in $G$. Then a.a.s. the following hold: (i) $G$ can be decomposed into $\lfloor\Delta(G)/2\rfloor$ cycles and a matching of size $odd(G)/2$. (ii) $G$ can be decomposed into $\max\{odd(G)/2,\lceil\Delta(G)/2\rceil\}$ paths. (iii) $G$ can be decomposed into $\lceil\Delta(G)/2\rceil$ linear forests. Each of these bounds is best possible. We actually derive (i)--(iii) from `quasirandom' versions of our results. In that context, we also determine the edge chromatic number of a given dense quasirandom graph of even order. For all these results, our main tool is a result on Hamilton decompositions of robust expanders by K\"uhn and Osthus.Comment: Some typos from the first version have been correcte

Spanning Trees and Spanning Eulerian Subgraphs with Small Degrees. II

Let $G$ be a connected graph with $X\subseteq V(G)$ and with the spanning forest $F$. Let $\lambda\in [0,1]$ be a real number and let $\eta:X\rightarrow (\lambda,\infty)$ be a real function. In this paper, we show that if for all $S\subseteq X$, $\omega(G\setminus S)\le\sum_{v\in S}\big(\eta(v)-2\big)+2-\lambda(e_G(S)+1)$, then $G$ has a spanning tree $T$ containing $F$ such that for each vertex $v\in X$, $d_T(v)\le \lceil\eta(v)-\lambda\rceil+\max\{0,d_F(v)-1\}$, where $\omega(G\setminus S)$ denotes the number of components of $G\setminus S$ and $e_G(S)$ denotes the number of edges of $G$ with both ends in $S$. This is an improvement of several results and the condition is best possible. Next, we also investigate an extension for this result and deduce that every $k$-edge-connected graph $G$ has a spanning subgraph $H$ containing $m$ edge-disjoint spanning trees such that for each vertex $v$, $d_H(v)\le \big\lceil \frac{m}{k}(d_G(v)-2m)\big\rceil+2m$, where $k\ge 2m$; also if $G$ contains $k$ edge-disjoint spanning trees, then $H$ can be found such that for each vertex $v$, $d_H(v)\le \big\lceil \frac{m}{k}(d_G(v)-m)\big\rceil+m$, where $k\ge m$. Finally, we show that strongly $2$-tough graphs, including $(3+1/2)$-tough graphs of order at least three, have spanning Eulerian subgraphs whose degrees lie in the set $\{2,4\}$. In addition, we show that every $1$-tough graph has spanning closed walk meeting each vertex at most $2$ times and prove a long-standing conjecture due to Jackson and Wormald (1990).Comment: 46 pages, Keywords: Spanning tree; spanning Eulerian; spanning closed walk; connected factor; toughness; total exces

Factors and Connected Factors in Tough Graphs with High Isolated Toughness

In this paper, we show that every $1$-tough graph with order and isolated toughness at least $r+1$ has a factor whose degrees are $r$, except for at most one vertex with degree $r+1$. Using this result, we conclude that every $3$-tough graph with order and isolated toughness at least $r+1$ has a connected factor whose degrees lie in the set $\{r,r+1\}$, where $r\ge 3$. Also, we show that this factor can be found $m$-tree-connected, when $G$ is a $(2m+\epsilon)$-tough graph with order and isolated toughness at least $r+1$, where $r\ge (2m-1)(2m/\epsilon+1)$ and $\epsilon > 0$. Next, we prove that every $(m+\epsilon)$-tough graph of order at least $2m$ with high enough isolated toughness admits an $m$-tree-connected factor with maximum degree at most $2m+1$. From this result, we derive that every $(2+\epsilon)$-tough graph of order at least three with high enough isolated toughness has a spanning Eulerian subgraph whose degrees lie in the set $\{2,4\}$. In addition, we provide a family of $5/3$-tough graphs with high enough isolated toughness having no connected even factors with bounded maximum degree

On some intriguing problems in Hamiltonian graph theory -- A survey

We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, $t$-tough graphs, and claw-free graphs