A labeling of a connected graph $G$ on $n$ vertices is a bijection $c:V(G)\to\{1,\dots,n\}$; writing $c(u,v)=|c(u)-c(v)|$, a vertex $u$ is list-distance consistent if $d(u,v)<d(u,w)$ implies $c(u,v)\le c(u,w)$ for all $v,w$. The maximum number of such vertices over all labelings is the list-distance consistency ldc$(G)$, introduced by Casselgren and Henricsson. We prove that in a tree, the consistent vertices of any labeling lie on a single path, along which the labels form a block of consecutive integers in increasing order (with respect to a suitable orientation of the path), no vertex off the path receiving a label from that block. We deduce that ldc equals $3$ for every complete $k$-ary tree except the binary tree of height two, and we determine ldc for all spiders.
A full answer to the problem of establishing if there exists a simple graph on v vertices having $\gamma$ as its co-degree sequence and, in case of positive answer, provide such a graph for the class of planar $C_4$-free graphs.
M. Ascolese, Pietro Negrini, S. Pagani et al.· 0 citations
Let $G(V,E)$ be a graph with $V(G)$ as the set of vertices and $E(G)$ as the set of edges. A labeling is a bijection $f:E(G)\to\{1,2,\cdots, |E(G)|\}$. For each vertex \(v\), let \(\phi(v)\) be the sum of the labels on the edges incident to \(v\). If all values \(\phi(v)\) are distinct, the labeling is antimagic, and t...
Poh-Hwa Ong, Huey Voon Chen, Wei-She-An Ng· Journal of the Indonesian Ma...· 0 citations
A graph $G$ is $n$-existentially closed or $n$-e.c. if, for all subsets $S\subseteq V(G)$ with $|S|=n$ and for all partitions $S=A\sqcup B$, there exists a vertex in $V(G)\setminus S$ adjacent to all vertices in $A$ and no vertices in $B$. We study the minimum number of edges $m(v,n)$ of a $v$-vertex $n$-e.c. graph, an...
Isabel T. Byrne, John Byrne, S. Cioabă· 0 citations
A set $S$ of vertices of a graph $G$ is a connected mutual-visibility set if every two vertices of $S$ are joined by a shortest path whose internal vertices lie outside $S$, and the subgraph induced by $S$ is connected. We introduce the connected mutual-visibility number $\mu_c(G)$, defined as the maximum cardinality o...
Let $D$ be a digraph, and let $w: E(D) \to \{1,2,\ldots \}$ be a positive integer weight assignment on the arcs of $D$. An arc $e=(u,v)$ is called dominating if $w(e) > d_{(D,w)}^+(v)$, where $d_{(D,w)}^+(v)$ denotes the weighted out-degree of $v$. We say $(D,w)$ is acyclic if every non-empty sub-digraph $(D', w)$ has...
An edge labeling of a connected graph $G=(V, E)$ is called a local antimagic labeling if it is a bijection $f: E\rightarrow \{1, 2, \cdots, \left | E \right | \}$ such that for any pair of adjacent vertices $u$ and $v$, $f^{+}(u)\neq f^{+}(v)$ where $f^{+}(x)=\sum f(e)$, with $e$ ranging over all the edges incident to...
