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Let $G$ be a graph and $g,f:V(G)\to2^N$ be two set functions such that $g(v)\le f(v)$ and $g(v)\equiv f(v)\pmod 2$ for every $v\in V(G)$. An orientation $O$ of $G$ is called a $(g,f)$-parity orientation if $g(v)\le d^+_O(v)\le f(v)$ and $g(v)\equiv d^+_O(v)\pmod 2$ for every $v\in V(G)$. In this paper, we give a Tutte-type characterization for a graph to have a $(g,f)$-parity orientation.

Let $G$ be a graph and $F:V(G)\to2^N$ be a set function. The graph $G$ is said to be \emph{F-avoiding} if there exists an orientation $O$ of $G$ such that $d^+_O(v)\notin F(v)$ for every $v\in V(G)$, where $d^+_O(v)$ denotes the out-degree of $v$ in the directed graph $G$ with respect to $O$. In this paper, we give a Tutte-type good characterization to decide the $F$-avoiding problem when for every $v\in V(G)$, $|F(v)|\leq \frac{1}{2}(d_G(v)+1)$ and $F(v)$ contains no two consecutive integers. Our proof also gives a simple polynomial algorithm to find a desired orientation. As a corollary, we prove the following result: if for every $v\in V(G)$, $|F(v)|\leq \frac{1}{2}(d_G(v)+1)$ and $F(v)$ contains no two consecutive integers, then $G$ is $F$-avoiding. This partly answers a problem proposed by Akbari et. al.(2020)

Let $n,m$ be integers such that $1\leq m\leq (n-2)/2$ and let $[n]=\{1,\ldots,n\}$. Let $\mathcal{G}=\{G_1,\ldots,G_{m+1}\}$ be a family of graphs on the same vertex set $[n]$. In this paper, we prove that if for any $i\in [m+1]$, the spectral radius of $G_i$ is not less than $\max\{2m,\frac{1}{2}(m-1+\sqrt{(m-1)^2+4m(n-m)})\}$, then $\mathcal{G}$ admits a rainbow matching, i.e. a choice of disjoint edges $e_i\in G_i$, unless $G_1=G_2=\ldots=G_{m+1}$ and $G_1\in \{K_{2m+1}\cup (n-2m-1)K_1, K_m\vee (n-m)K_1\}$.

MAX phase family has been extended by the addition of late transition metals at the A-site with the expectation of diverse functional properties, such as magnetism and catalysis. Here, we present our systematic density functional investigation on the phase stability and physical properties of newly synthesized Ti2ZnX (X = C, N) phasesin comparison with conventional Ti2AlX (X = C, N).Due to smaller size of N as compared to C, the unit cell dimensionis reduced when C atoms are replaced by N atoms atthe X-site. The thermodynamic, mechanical and dynamical stabilities are validatedby estimating the formation energies, elastic constants and phonon dispersions, respectively. The elastic properties of Ti2ZnN are nearly isotropic while those of Ti2ZnC are completely anisotropic. To understand the thin-film characteristicsin Ti2ZnX, the surface properties with (001)-terminated slabs are investigated. Both Ti2ZnX bulk and (001)-surfaces exhibit metal-like electronic structure. There is a strong covalent bonding between Ti-X and Ti-Zn atoms.Additional states are generated at the Fermi level (EF) due to the unusual d-pstates hybridization between Ti and Zn atoms.The anisotropy in chemical bonding is confirmed by the cleavage energy difference between Ti-X and Ti-Zn atoms. Here,Ti(X)-001 and Zn-001 terminations are stable surfaces, however, in terms of chemical potentials, Zn-001 termination is the most favorable.