Spanning Eulerian subgraphs and Catlin’s reduced graphs

A graph G is collapsible if for every even subset R ⊆ V (G), there is a spanning connected subgraph HR of G whose set of odd degree vertices is R. A graph is reduced if it has no nontrivial collapsible subgraphs. Catlin [4] showed that the existence of spanning Eulerian subgraphs in a graph G can be determined by the reduced graph obtained from G by contracting all the collapsible subgraphs of G. In this paper, we present a result on 3-edge-connected reduced graphs of small orders. Then, we prove that a 3-edge-connected graph G of order n either has a spanning Eulerian subgraph or can be contracted to the Petersen graph if G satisfies one of the following: (i) d(u) + d(v) \u3e 2(n/15 − 1) for any uv 6∈ E(G) and n is large; (ii) the size of a maximum matching in G is at most 6; (iii) the independence number of G is at most 5. These are improvements of prior results in [16], [18], [24] and [25]

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 $15$

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

On minimum degree conditions for supereulerian graphs

A graph is called supereulerian if it has a spanning closed trail. Let $G$ be a 2-edge-connected graph of order $n$ such that each minimal edge cut $E \subseteq E (G)$ with $|E| \le 3$ satisfies the property that each component of $G-E$ has order at least $(n-2)/5$. We prove that either $G$ is supereulerian or $G$ belongs to one of two classes of exceptional graphs. Our results slightly improve earlier results of Catlin and Li. Furthermore our main result implies the following strengthening of a theorem of Lai within the class of graphs with minimum degree $\delta\ge 4$: If $G$ is a 2-edge-connected graph of order $n$ with $\delta (G)\ge 4$ such that for every edge $xy\in E (G)$ , we have $\max \{d(x),d(y)\} \ge (n-7)/5$, then either $G$ is supereulerian or $G$ belongs to one of two classes of exceptional graphs. We show that the condition $\delta(G)\ge 4$ cannot be relaxed

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