Supereulerian graphs and the Petersen graph

A graphG is supereulerian if G has a spanning eulerian subgraph.Boesch et al.[J.Graph Theory,1,79–84(1977)]proposed the problem of characterizing supereulerian graphs.In this paper,we prove that any 3-edge-connected graph with at most 11 edge-cuts of size 3 is supereulerian if and only if it cannot be contractible to the Petersen graph.This extends a former result of Catlin and Lai[J.Combin.Theory,Ser.B,66,123–139(1996)].     

Nowhere-zero 15-flow in 3-edge-connected bidirected graphs

It was conjectured by Bouchet that every bidirected graph which admits a nowhere-zero κ flow will admit a nowhere-zero 6-flow. He proved that the conjecture is true when 6 is replaced by 216. Zyka improved the result with 6 replaced by 30. Xu and Zhang showed that the conjecture is true for 6-edge-connected graphs. And for 4-edge-connected graphs, Raspaud and Zhu proved it is true with 6 replaced by 4. In this paper, we show that Bouchet＇s conjecture is true with 6 replaced by 15 for 3-edge-connected graphs.     

Solutions of the Cauchy problem for the 2-D Euler system with fan-shaped structure

We study the Cauchy problem for the isentropic 2-D Euler system with initial data having discontinuity on a smooth curve. A local existence of a solution is established, which consists of shock wave, rarefaction wave and contact discontinuity.     

Cropper's question and Cruse's theorem about partial Latin squares

In 1974 Cruse gave necessary and sufficient conditions for an r × s partial latin square P on symbols σ₁,σ₂,…,σ_t, which may have some unfilled cells, to be completable to an n × n latin square on symbols σ₁,σ₂,…,σ_n, subject to the condition that the unfilled cells of P must be filled with symbols chosen from {σ_{t + 1},σ_{t + 2},…,σ_n}. These conditions consisted of r + s + t + 1 inequalities. Hall's condition applied to partial latin squares is a necessary condition for their completion, and is a generalization of, and in the spirit of Hall's Condition for a system of distinct representatives. Cropper asked whether Hall's Condition might also be sufficient for the completion of partial latin squares, but we give here a counterexample to Cropper's speculation. We also show that the r + s + t + 1 inequalities of Cruse's Theorem may be replaced by just four inequalities, two of which are Hall inequalities for P (i.e. two of the inequalities which constitute Hall's Condition for P), and the other two are Hall inequalities for the conjugates of P. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:268‐279, 2011     

Some applications of the AMPC model of the shear yield stress of particulate fluids

The AMPC structural model of shear yield stress of single component suspensions is extended to three important rheological phenomena encountered in particulate fluids. These are shear yield stress of multi-component systems, time dependent shear yield stress of thixotropic suspensions, and compressive yield stress of particulate fluids. The shear yield stress of multi-component suspensions is modeled by summing the interparticle interactions due to all possible combinations of particle sizes and solid constituents. The time dependent shear yield stress of thixotropic muds is simulated by invoking the proportionality between yield stress and interparticle bond density across the shear plane. The compressive yield stress of particulate fluids is calculated by assuming that consolidation occurs by shear but at a narrower gap between the particles. The ability of the model to describe these diverse phenomena reasonably well seemingly validates its basic premise, namely, a 3-dimensional space-filling network of particles whose mechanical properties can be estimated from the theory of strength of particulate assemblage.     

有向跳图的生成欧拉子有向图

一个有向多重图D的跳图$J(D)$是一个顶点集为$D$的弧集,其中$(a,b)$是$J(D)$的一条弧当且仅当存在有向多重图$D$中的顶点$u_1$, $v_1$, $u_2$, $v_2$,使得$a=(u_1,v_1)$, $b=(u_2,v_2)$ 并且$v_1\neq u_2$.本文刻画了有向多重图类$\mathcal{H}_1$和$\mathcal{H}_2$,并证明了一个有向多重图$D$的跳图$J(D)$是强连通的当且仅当$D\not\in \mathcal{H}_1$.特别地, $J(D)$是弱连通的当且仅当$D\not\in \mathcal{H}_2$.进一步, 得到以下结果: (i) 存在有向多重图类$\mathcal{D}$使得有向多重图$D$的强连通跳图$J(D)$是强迹连通的当且仅当$D\not\in\mathcal{D}$. (ii) 每一个有向多重图$D$的强连通跳图$J(D)$是弱迹连通的,因此是超欧拉的. (iii) 每一个有向多重图D的弱连通跳图$J(D)$含有生成迹.     

Supereulerian regular matroids without small cocircuits

A cycle of a matroid is a disjoint union of circuits. A matroid is supereulerian if it contains a spanning cycle. To answer an open problem of Bauer in 1985, Catlin proved in [J. Graph Theory 12 (1988) 29–44] that for sufficiently large $n$, every 2-edge-connected simple graph $G$ with $\text{◂=▸}n=\mid V\left(G\right)\mid$ and minimum degree $\text{◂≥▸}\delta \left(G\right)\ge \frac{n}{5}$ is supereulerian. In [Eur. J. Combinatorics, 33 (2012), 1765–1776], it is shown that for any connected simple regular matroid $M$, if every cocircuit $D$ of $M$ satisfies $\text{◂≥▸}\mid D\mid \ge \max \left\{\frac{\text{◂−▸}r\left(M\right)-5}{5},6\right\}$, then $M$ is supereulerian. We prove the following. (i) Let $M$ be a connected simple regular matroid. If every cocircuit $D$ of $M$ satisfies $\text{◂≥▸}\mid D\mid \ge \max \left\{\frac{\text{◂+▸}r\left(M\right)+1}{10},9\right\}$, then $M$ is supereulerian. (ii) For any real number $c$ with , there exists an integer $f\left(c\right)$ such that if every cocircuit $D$ of a connected simple cographic matroid $M$ satisfies $\mid D\mid \ge \text{◂lim▸}\max \text{◂{}▸}\{c\text{◂()▸}\left(r\left(M\right)+1\right),f\left(c\right)\}$, then $M$ is supereulerian.