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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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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.
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.
Suppose that the vertices of a graph $G$ are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the majority color (if it exists), and any vertex of this color is called a majority vertex. We study the problem of finding a majority vertex (or show that none exists) if we can query edges to learn whether their endpoints have the same or different colors. Denote the least number of queries needed in the worst case by $m(G)$. It was shown by Saks and Werman that $m(K_n)=n-b(n)$, where $b(n)$ is the number of 1s in the binary representation of $n$. In this paper, we initiate the study of the problem for general graphs. The obvious bounds for a connected graph $G$ on $n$ vertices are $n-b(n)le m(G)le n-1$. We show that for any tree $T$ on an even number of vertices we have $m(T)=n-1$ and that for any tree $T$ on an odd number of vertices, we have $n-65le m(T)le n-2$. Our proof uses results about the weighted version of the problem for $K_n$, which may be of independent interest. We also exhibit a sequence $G_n$ of graphs with $m(G_n)=n-b(n)$ such that $G_n$ has $O(nb(n))$ edges and $n$ vertices.
Motivated by analogous questions in the setting of Steiner triple systems and Latin squares, Nenadov, Sudakov and Wagner [Completion and deficiency problems, Journal of Combinatorial Theory Series B, 2020] recently introduced the notion of graph deficiency. Given a global spanning property $mathcal P$ and a graph $G$, the deficiency $text{def}(G)$ of the graph $G$ with respect to the property $mathcal P$ is the smallest non-negative integer $t$ such that the join $G*K_t$ has property $mathcal P$. In particular, Nenadov, Sudakov and Wagner raised the question of determining how many edges an $n$-vertex graph $G$ needs to ensure $G*K_t$ contains a $K_r$-factor (for any fixed $rgeq 3$). In this paper we resolve their problem fully. We also give an analogous result which forces $G*K_t$ to contain any fixed bipartite $(n+t)$-vertex graph of bounded degree and small bandwidth.
We introduce the notion of a properly ordered coloring (POC) of a weighted graph, that generalizes the notion of vertex coloring of a graph. Under a POC, if $xy$ is an edge, then the larger weighted vertex receives a larger color; in the case of equal weights of $x$ and $y$, their colors must be different. In this paper, we shall initiate the study of this special coloring in graphs. For a graph $G$, we introduce the function $f(G)$ which gives the maximum number of colors required by a POC over all weightings of $G$. We show that $f(G)=ell(G)$, where $ell(G)$ is the number of vertices of a longest path in $G$. Another function we introduce is $chi_{POC}(G;t)$ giving the minimum number of colors required over all weightings of $G$ using $t$ distinct weights. We show that the ratio of $chi_{POC}(G;t)-1$ to $chi(G)-1$ can be bounded by $t$ for any graph $G$; in fact, the result is shown by determining $chi_{POC}(G;t)$ when $G$ is a complete multipartite graph. We also determine the minimum number of colors to give a POC on a vertex-weighted graph in terms of the number of vertices of a longest directed path in an orientation of the underlying graph. This extends the so called Gallai-Hasse-Roy-Vitaver theorem, a classical result concerning the relationship between the chromatic number of a graph $G$ and the number of vertices of a longest directed path in an orientation of $G$.
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