Let G be a 2-connected simple graph on sufficiently large n vertices. Catlin proved that if δ ( G ) ≥ n 5 , then G is supereulerian. Let M be a connected simple regular matroid. Huo et al. proved that M is supereulerian if g ∗ ( M ) ≥ max { r ( M ) + 1 10 , 9 } . In this paper, we prove that a simple connected regular matroid M is supereulerian if g ∗ ( M ) ≥ max { r ( M ) + 1 10 , 8 } , or g ∗ ( M ) ≥ max { r ( M ) + 1 15 , 9 } .

Abelian group colorings were first introduced by Jaeger et al. in [ J. Combin. Theory Ser. B, 56 (1992), pp. 165--182] as the dual concept of group connectivity of graphs. For given groups $\Gamma_1$ and $\Gamma_2$ with $|\Gamma_1| = |\Gamma_2|$, the dual version of a problem raised by Jaeger et al. suggests to investigate whether every $\Gamma_1$-colorable graph $G$ is also $\Gamma_2$-colorable. Recently, Hǔsek, Mohelníková, and Šámal [ J. Graph Theory, 93 (2019), pp. 317--327] used computer testing to find the first examples of $\mathbb{Z}_4$-connected but not $\mathbb{Z}_2^2$-connected graphs as well as $\mathbb{Z}_2^2$-connected but not $\mathbb{Z}_4$-connected graphs. As their examples are nonplanar, the group coloring problem remains unanswered. Group coloring was extended to non-abelian groups in Li and Lai [ Discrete Math., 313 (2013), pp. 101--104]. We introduce a group coloring local structure (defined as a snarl in the paper) and use it to construct infinitely many ordered triples $(G, \Gamma_1, \Gamma_2)$ in which $G$ is a graph and $\Gamma_1$ and $\Gamma_2$ are groups with $|\Gamma_1| = |\Gamma_2|$, such that $G$ is $\Gamma_1$-colorable but not $\Gamma_2$-colorable.

A mod $(2p+1)$-orientation $D$ is an orientation of $G$ such that $d_D^+(v)-d_D^-(v)\equiv 0 \pmod {2p+1}$ for any vertex $v \in V(G)$. Jaeger conjectured that every $4p$-edge-connected graph has a mod $(2p+1)$-orientation. A graph $G$ is strongly ${\mathbb Z}_{2p+1}$-connected if for every mapping $b: V(G) \mapsto {\mathbb Z}_{2p+1}$ with $\sum_{v\in V(G)}b(v)=0$, there exists an orientation $D$ of $G$ such that $d_D^+(v)-d_D^-(v)= b(v)$ in ${\mathbb Z}_{2p+1}$ for any $v \in V(G)$. A strongly ${\mathbb Z}_{2p+1}$-connected graph admits a mod $(2p+1)$-orientation, and it is a contractible configuration for mod $(2p+1)$-orientation. We prove Jaeger's module orientation conjecture is equivalent to its restriction to bipartite simple graphs and investigate strongly ${\mathbb Z}_{2p+1}$-connectedness of certain bipartite graphs, particularly for $p=2$. We also show that if $G$ is a simple graph with $|V(G)|\ge N(p)= 1152p^4$ and $\min\{\delta(G),\delta(G^c)\}\ge 4p$, then either $G$ or $G^c$ is strongly ${\mathbb Z}_{2p+1}$-connected. When $p=2$, the value of $N(2)$ can be reduced to $N(2) = 80$.

Reliability analysis of an interconnection network is of great significance to the design and maintenance of multiprocessor systems. The h-extra edge-connectivity of a given interconnected network G with N processors, denoted by λh(G), is the minimum cardinality of set of faulty links, such that whose removal will disconnect the network with all its resulting components having at least h processors for h ≤ N/2. It gives a more refined quantitative analysis of indicators of the robustness of a multiprocessor system in the presence of failing links. The n-dimensional folded hypercube FQn, as one of potential interconnected networks, is a well-known variation of the hypercube structure with N = 2nprocessors. In this paper, the h-extra edge-connectivity of the network FQn, λh(FQn), is first investigated for each well-defined positive integer h ≤ N/2. We divide the interval 1 ≤ h ≤ N/2 into some subintervals and obtain some properties of λh(FQn) in these subintervals. Then, we deduce a recursive relation of λh(F Qn). Based on this recursion, an efficient O(log2(N)) algorithm is designed to totally determine the exact values and λh-optimality of λh(FQn) for each h ≤ N/2.

Let $\mathbb Z_m$ be the cyclic group of order $m \geq 3$. A graph $G$ is $\mathbb Z_m$-connected if $G$ has an orientation $D$ such that for any mapping $b: V(G) \mapsto \mathbb Z_m$ with $\sum_{v\in V(G)}b(v)=0$, there exists a mapping $f:E(G) \mapsto \mathbb Z_m -\{0\}$ satisfying $\sum_{e\in E_D^+(v)} f(e) - \sum_{e\in E_D^-(v)} f(e) = b(v)$ in $\mathbb Z_m$ for any $v \in V(G)$; and a graph $G$ is strongly $\mathbb Z_m$-connected if, for any mapping $\theta: V(G)\rightarrow \mathbb Z_m$ with $\sum_{v\in V(G)}\theta(v) = |E(G)|$ in $\mathbb Z_m$, there is an orientation $D$ such that $d_D^+(v)=\theta(v)$ in $\mathbb Z_m$ for each $v \in V(G)$. In this paper, we study the relation between $\mathbb Z_m$-connected graphs and strongly $\mathbb Z_m$-connected graphs and show that a graph $G$ is $\mathbb Z_m$-connected if and only if $(m-2)G$ is strongly $\mathbb Z_m$-connected, where $(m-2)G$ is the graph obtained from $G$ by replacing each edge in $G$ with $m-2$ parallel edges. We also show that if $G$ is $\mathbb Z_m$-connected, then $(m-2)G$ has $m-1$ edge disjoint spanning trees. Those results together with a result by Jaeger et al. [J. Combin. Theory Ser. B, 56 (1992), pp. 165-182] imply that every $\mathbb Z_3$-connected graph is $A$-connected for any abelian group $A$ with $|A| \geq 4$. They are applied to determine the exact values of $ex(n,\mathbb Z_m)$ for all $m\geq 3$, where $ex(n,\mathbb Z_m)$ is the largest integer such that every simple graph on $n$ vertices with at most $ex(n,\mathbb Z_m)$ edges is not $\mathbb Z_m$-connected, and to present characterizations of graphic and multigraphic sequences that have $\mathbb Z_m$-connected realizations.

An orientation of a graph $G$ is a mod (2p+1)-orientation if, under this orientation, the net out-degree at every vertex is congruent to zero mod 2p+1. If, for any function $b: V(G) \rightarrow \mathbb Z_{2p+1}$ satisfying $\sum_{v \in V(G)} b(v) \equiv 0$ (mod 2p+1), $G$ always has an orientation $D$ such that the net out-degree at every vertex $v$ is congruent to $b(v)$ mod 2p+1, then $G$ is strongly $\mathbb Z_{2p+1}$-connected. The graph $G'$ obtained from $G$ by contracting all nontrivial subgraphs that are strongly $\mathbb Z_{2s+1}$-connected is called the $\mathbb Z_{2s+1}$-reduction of $G$. Motivated by a minimum degree condition of Barat and Thomassen [J. Graph Theory, 52 (2006), pp. 135--146], and by the Ore conditions of Fan and Zhou [SIAM J. Discrete Math., 22 (2008), pp. 288--294] and of Luo et al. [European J. Combin., 29 (2008), pp. 1587--1595] on $\mathbb Z_3$-connected graphs, we prove that for a simple graph $G$ on $n$ vertices, and for any integers $s > 0$ and real numbers $\alpha, \beta$ with $0 < \alpha < 1$, if for any nonadjacent vertices $u, v \in V(G)$, $d_G(u) + d_G(v) \ge \alpha n + \beta$, then there exists a finite family ${\cal {F}}(\alpha,s)$ of nonstrongly $\mathbb Z_{2s+1}$-connected graphs such that either $G$ is strongly $\mathbb Z_{2s+1}$-connected or the $\mathbb Z_{2s+1}$-reduction of $G$ is in ${\cal {F}}(\alpha,s)$.

True steady states are a rare occurrence in living organisms, yet their knowledge is essential for quasi-steady-state approximations, multistability analysis, and other important tools in the investigation of chemical reaction networks (CRN) used to describe molecular processes on the cellular level. Here, we present an approach that can provide closed form steady-state solutions to complex systems, resulting from CRN with binary reactions and mass-action rate laws. We map the nonlinear algebraic problem of finding steady states onto a linear problem in a higher-dimensional space. We show that the linearized version of the steady-state equations obeys the linear conservation laws of the original CRN. We identify two classes of problems for which complete, minimally parameterized solutions may be obtained using only the machinery of linear systems and a judicious choice of the variables used as free parameters. We exemplify our method, providing explicit formulae, on CRN describing signal initiation of two important types of RTK receptor-ligand systems, VEGF and EGF-ErbB1.

In this paper, we establish an equivalence between the contractible graphs with respect to the mod $(2p+1)$-orientability and the graphs with $K_{1, 2p+1}$-decompositions. This is applied to disprove a conjecture proposed by Barat and Thomassen that every 4-edge-connected simple planar graph G with $|E(G)|\equiv 0$ (mod 3) has a claw decomposition.