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A graph $G$ is $k$-$weighted-list-antimagic$ if for any vertex weighting $omegacolon V(G)tomathbb{R}$ and any list assignment $Lcolon E(G)to2^{mathbb{R}}$ with $|L(e)|geq |E(G)|+k$ there exists an edge labeling $f$ such that $f(e)in L(e)$ for all $ein E(G)$, labels of edges are pairwise distinct, and the sum of the labels on edges incident to a vertex plus the weight of that vertex is distinct from the sum at every other vertex. In this paper we prove that every graph on $n$ vertices having no $K_1$ or $K_2$ component is $lfloor{frac{4n}{3}}rfloor$-weighted-list-antimagic. An oriented graph $G$ is $k$-$oriented-antimagic$ if there exists an injective edge labeling from $E(G)$ into ${1,dotsc,|E(G)|+k}$ such that the sum of the labels on edges incident to and oriented toward a vertex minus the sum of the labels on edges incident to and oriented away from that vertex is distinct from the difference of sums at every other vertex. We prove that every graph on $n$ vertices with no $K_1$ component admits an orientation that is $lfloor{frac{2n}{3}}rfloor$-oriented-antimagic.
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Let $G = (V, E)$ be a finite simple undirected graph without $K_2$ components. A bijection $f : E rightarrow {1, 2,cdots, |E|}$ is called a {bf local antimagic labeling} if for any two adjacent vertices $u$ and $v$, they have different vertex sums, i.e. $w(u) eq w(v)$, where the vertex sum $w(u) = sum_{e in E(u)} f(e)$, and $E(u)$ is the set of edges incident to $u$. Thus any local antimagic labeling induces a proper vertex coloring of $G$ where the vertex $v$ is assigned the color(vertex sum) $w(v)$. The {bf local antimagic chromatic number} $chi_{la}(G)$ is the minimum number of colors taken over all colorings induced by local antimagic labelings of $G$. In this article among others we determine completely the local antimagic chromatic number $chi_{la}(Gcirc overline{K_m})$ for the corona product of a graph $G$ with the null graph $overline{K_m}$ on $mgeq 1$ vertices, when $G$ is a path $P_n$, a cycle $C_n$, and a complete graph $K_n$.
Let $ G $ be a simple graph of $ ell $ vertices $ {1, dots, ell } $ with edge set $ E_{G} $. The graphical arrangement $ mathcal{A}_{G} $ consists of hyperplanes $ {x_{i}-x_{j}=0} $, where $ {i, j } in E_{G} $. It is well known that three properties, chordality of $ G $, supersolvability of $ mathcal{A}_{G} $, and freeness of $ mathcal{A}_{G} $ are equivalent. Recently, Richard P. Stanley introduced $ psi $-graphical arrangement $ mathcal{A}_{G, psi} $ as a generalization of graphical arrangements. Lili Mu and Stanley characterized the supersolvability of the $ psi $-graphical arrangements and conjectured that the freeness and the supersolvability of $ psi $-graphical arrangements are equivalent. In this paper, we will prove the conjecture.
Given a graph $G=(V,E)$ and a colouring $f:Emapsto mathbb N$, the induced colour of a vertex $v$ is the sum of the colours at the edges incident with $v$. If all the induced colours of vertices of $G$ are distinct, the colouring is called antimagic. If $G$ has a bijective antimagic colouring $f:Emapsto {1,dots,|E|}$, the graph $G$ is called antimagic. A conjecture of Hartsfield and Ringel states that all connected graphs other than $K_2$ are antimagic. Alon, Kaplan, Lev, Roddity and Yuster proved this conjecture for graphs with minimum degree at least $c log |V|$ for some constant $c$; we improve on this result, proving the conjecture for graphs with average degree at least some constant $d_0$.
A $labeling$ of a digraph $D$ with $m$ arcs is a bijection from the set of arcs of $D$ to ${1,2,ldots,m}$. A labeling of $D$ is $antimagic$ if no two vertices in $D$ have the same vertex-sum, where the vertex-sum of a vertex $u in V(D)$ for a labeling is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. An antimagic orientation $D$ of a graph $G$ is $antimagic$ if $D$ has an antimagic labeling. Hefetz, M$ddot{u}$tze and Schwartz in [J. Graph Theory 64(2010)219-232] raised the question: Does every graph admits an antimagic orientation? It had been proved that for any integer $d$, every 2$d$-regular graph with at most two odd components has an antimagic orientation. In this paper, we consider the 2$d$-regular graph with many odd components. We show that every 2$d$-regular graph with any odd components has an antimagic orientation provide each odd component with enough order.
Let $G$ be a graph, and let $w$ be a positive real-valued weight function on $V(G)$. For every subset $S$ of $V(G)$, let $w(S)=sum_{v in S} w(v).$ A non-empty subset $S subset V(G)$ is a weighted safe set of $(G,w)$ if, for every component $C$ of the subgraph induced by $S$ and every component $D$ of $G-S$, we have $w(C) geq w(D)$ whenever there is an edge between $C$ and $D$. If the subgraph of $G$ induced by a weighted safe set $S$ is connected, then the set $S$ is called a connected weighted safe set of $(G,w)$. The weighted safe number $mathrm{s}(G,w)$ and connected weighted safe number $mathrm{cs}(G,w)$ of $(G,w)$ are the minimum weights $w(S)$ among all weighted safe sets and all connected weighted safe sets of $(G,w)$, respectively. Note that for every pair $(G,w)$, $mathrm{s}(G,w) le mathrm{cs}(G,w)$ by their definitions. Recently, it was asked which pair $(G,w)$ satisfies the equality and shown that every weighted cycle satisfies the equality. In this paper, we give a complete list of connected bipartite graphs $G$ such that $mathrm{s}(G,w)=mathrm{cs}(G,w)$ for every weight function $w$ on $V(G)$.
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