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In this paper we present some results on the global existence of weak solutions to a nonlinear variational wave equation and some related problems. We first introduce the main tools, the $L^p$ Young measure theory and related compactness results, in the first section. Then we use the $L^p$ Young measure theory to prove the global existence of dissipative weak solutions to the asymptotic equation of the nonlinear wave equation, and comment on its relation to Camassa-Holm equations in the second section....

For a nonempty set $S$ of vertices in a strong digraph $D$, the strong distance $d\left(S\right)$ is the minimum size of a strong subdigraph of $D$ containing the vertices of $S$. If $S$ contains $k$ vertices, then $d\left(S\right)$ is referred to as the $k$-strong distance of $S$. For an integer $k\ge 2$ and a vertex $v$ of a strong digraph $D$, the $k$-strong eccentricity ${\mathrm{s}e}_k\left(v\right)$ of $v$ is the maximum $k$-strong distance $d\left(S\right)$ among all sets $S$ of $k$ vertices in $D$ containing $v$. The minimum $k$-strong eccentricity among the vertices of $D$ is its $k$-strong radius ${\mathrm{s}rad}_{k}D$ and the maximum...

For integers k and n with 2 ≤ k ≤ n − 1, a graph G of order n is k-path pancyclic if every path P of order k in G lies on a cycle of every length from k + 1 to n. Thus a 2-path pancyclic graph is edge-pancyclic. In this paper, we present sufficient conditions for graphs to be k-path pancyclic. For a graph G of order n ≥ 3, we establish sharp lower bounds in terms of n and k for (a) the minimum degree of G, (b) the minimum degree-sum of nonadjacent vertices of G and (c) the size of G such that G...

Two vertices u and v in a nontrivial connected graph G are twins if u and v have the same neighbors in V (G) − {u, v}. If u and v are adjacent, they are referred to as true twins; while if u and v are nonadjacent, they are false twins. For a positive integer k, let c : V (G) → Zk be a vertex coloring where adjacent vertices may be assigned the same color. The coloring c induces another vertex coloring c′ : V (G) → Zk defined by c′(v) = P u∈N[v] c(u) for each v ∈ V (G), where N[v] is the closed neighborhood...

We show that for every integer k ≥ 2 and every k graphs G₁,G₂,...,Gₖ, there exists a hull graph with k hull vertices v₁,v₂,...,vₖ such that link $L\left(v_i\right)=G_i$ for 1 ≤ i ≤ k. Moreover, every pair a, b of integers with 2 ≤ a ≤ b is realizable as the hull number and geodetic number (or upper geodetic number) of a hull graph. We also show that every pair a,b of integers with a ≥ 2 and b ≥ 0 is realizable as the hull number and forcing geodetic number of a hull graph.

For two vertices u and v of a graph G, the set I(u, v) consists of all vertices lying on some u-v geodesic in G. If S is a set of vertices of G, then I(S) is the union of all sets I(u,v) for u, v ∈ S. A set S is a geodetic set if I(S) = V(G). A minimum geodetic set is a geodetic set of minimum cardinality and this cardinality is the geodetic number g(G). A subset T of a minimum geodetic set S is called a forcing subset for S if S is the unique minimum geodetic set containing T. The forcing geodetic...

A graph G is 2-stratified if its vertex set is partitioned into two classes (each of which is a stratum or a color class), where the vertices in one class are colored red and those in the other class are colored blue. Let F be a 2-stratified graph rooted at some blue vertex v. An F-coloring of a graph is a red-blue coloring of the vertices of G in which every blue vertex v belongs to a copy of F rooted at v. The F-domination number ${\gamma }_F\left(G\right)$ is the minimum number of red vertices in an F-coloring of G. In...

For a vertex v of a connected graph G and a subset S of V(G), the distance between v and S is d(v,S) = mind(v,x)|x ∈ S. For an ordered k-partition Π = S₁,S₂,...,Sₖ of V(G), the representation of v with respect to Π is the k-vector r(v|Π) = (d(v,S₁), d(v,S₂),..., d(v,Sₖ)). The k-partition Π is a resolving partition if the k-vectors r(v|Π), v ∈ V(G), are distinct. The minimum k for which there is a resolving k-partition of V(G) is the partition dimension pd(G) of G. A resolving partition Π = S₁,S₂,...,Sₖ...

For a connected graph G of order n ≥ 3, let f: E(G) → ℤₙ be an edge labeling of G. The vertex labeling f’: V(G) → ℤₙ induced by f is defined as $f^{\text{'}}\left(u\right)={\sum }_{v\in N\left(u\right)}f\left(uv\right)$, where the sum is computed in ℤₙ. If f’ is one-to-one, then f is called a modular edge-graceful labeling and G is a modular edge-graceful graph. A modular edge-graceful labeling f of G is nowhere-zero if f(e) ≠ 0 for all e ∈ E(G) and in this case, G is a nowhere-zero modular edge-graceful graph. It is shown that a connected graph G of order n ≥ 3 is nowhere-zero...

In this paper, we study the semiclassical limit of the cubic nonlinear Schrödinger equation with the Neumann boundary condition in an exterior domain. We prove that before the formation of singularities in the limit system, the quantum density and the quantum momentum converge to the unique solution of the compressible Euler equation with the slip boundary condition as the scaling parameter approaches $0.$

A graph $G$ is degree-continuous if the degrees of every two adjacent vertices of $G$ differ by at most 1. A finite nonempty set $S$ of integers is convex if $k\in S$ for every integer $k$ with $min\left(S\right)\le k\le max\left(S\right)$. It is shown that for all integers $r>0$ and $s\ge 0$ and a convex set $S$ with $min\left(S\right)=r$ and $max\left(S\right)=r+s$, there exists a connected degree-continuous graph $G$ with the degree set $S$ and diameter $2s+2$. The minimum order of a degree-continuous graph with a prescribed degree set is studied. Furthermore, it is shown that for every graph $G$ and convex set $S$ of...

For two vertices $u$ and $v$ of a graph $G$, the closed interval $I[u,v]$ consists of $u$, $v$, and all vertices lying in some $u\text{--}v$ geodesic of $G$, while for $S\subseteq V\left(G\right)$, the set $I\left[S\right]$ is the union of all sets $I[u,v]$ for $u,v\in S$. A set $S$ of vertices of $G$ for which $I\left[S\right]=V\left(G\right)$ is a geodetic set for $G$, and the minimum cardinality of a geodetic set is the geodetic number $g\left(G\right)$. A vertex $v$ in $G$ is an extreme vertex if the subgraph induced by its neighborhood is complete. The number of extreme vertices in $G$ is its extreme order $\mathrm{e}x\left(G\right)$. A graph $G$ is an extreme geodesic...

For an ordered set $W=\{w_1,w_2,\cdots ,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the representation of $v$ with respect to $W$ is the $k$-vector $r\left(v\right|W)$ = ($d(v,w_1)$, $d(v,w_2),\cdots ,d(v,w_k))$, where $d(x,y)$ represents the distance between the vertices $x$ and $y$. The set $W$ is a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. A resolving set for $G$ containing a minimum number of vertices is a basis for $G$. The dimension $dim\left(G\right)$ is the number of vertices in a basis for $G$. A resolving set $W$ of $G$ is connected if the subgraph...

For an ordered $k$-decomposition $𝒟=\{G_1,G_2,\cdots ,G_k\}$ of a connected graph $G$ and an edge $e$ of $G$, the $𝒟$-code of $e$ is the $k$-tuple $c_{𝒟}\left(e\right)=(d(e,G_1),d(e,G_2),...,d(e,G_k))$, where $d(e,G_i)$ is the distance from $e$ to $G_i$. A decomposition $𝒟$ is resolving if every two distinct edges of $G$ have distinct $𝒟$-codes. The minimum $k$ for which $G$ has a resolving $k$-decomposition is its decomposition dimension ${dim}_d\left(G\right)$. A resolving decomposition $𝒟$ of $G$ is connected if each $G_i$ is connected for $1\le i\le k$. The minimum $k$ for which $G$ has a connected resolving $k$-decomposition is its connected decomposition...

For two vertices $u$ and $v$ of a connected graph $G$, the set $I(u,v)$ consists of all those vertices lying on a $u$–$v$ geodesic in $G$. For a set $S$ of vertices of $G$, the union of all sets $I(u,v)$ for $u,v\in S$ is denoted by $I\left(S\right)$. A set $S$ is a convex set if $I\left(S\right)=S$. The convexity number $\mathrm{c}on\left(G\right)$ of $G$ is the maximum cardinality of a proper convex set of $G$. A convex set $S$ in $G$ with $\left|S\right|=\mathrm{c}on\left(G\right)$ is called a maximum convex set. A subset $T$ of a maximum convex set $S$ of a connected graph $G$ is called a forcing subset for $S$ if $S$ is the unique maximum convex set...

A proper coloring $c:V\left(G\right)\to \{1,2,...,k\}$, $k\ge 2$ of a graph $G$ is called a graceful $k$-coloring if the induced edge coloring $c^{\text{'}}:E\left(G\right)\to \{1,2,...,k-1\}$ defined by $c^{\text{'}}\left(uv\right)=|c\left(u\right)-c\left(v\right)|$ for each edge $uv$ of $G$ is also proper. The minimum integer $k$ for which $G$ has a graceful $k$-coloring is the graceful chromatic number ${\chi }_g\left(G\right)$. It is known that if $T$ is a tree with maximum degree $\Delta$, then ${\chi }_g\left(T\right)\le \lceil \frac{5}{3}\Delta \rceil$ and this bound is best possible. It is shown for each integer $\Delta \ge 2$ that there is an infinite class of trees $T$ with maximum degree $\Delta$ such that ${\chi }_g\left(T\right)=\lceil \frac{5}{3}\Delta \rceil$. In particular, we investigate for each integer $\Delta \ge 2$ a...

For an ordered set $W=\{w_1,w_2,\cdots ,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the (metric) representation of $v$ with respect to $W$ is the $k$-vector $r\left(v\right|W)$ = ($d(v,w_1),d(v,w_2),\cdots ,d(v,w_k)$), where $d(x,y)$ represents the distance between the vertices $x$ and $y$. The set $W$ is a resolving set for $G$ if distinct vertices of $G$ have distinct representations. A resolving set of minimum cardinality is a basis for $G$ and the number of vertices in a basis is its (metric) dimension $dim\left(G\right)$. For a basis $W$ of $G$, a subset $S$ of $W$ is called a forcing subset of $W$ if $W$ is...