An antimagic labelling of a graph G = (V,E) is a bijection from E to {1,2,...,|E|}, such that all vertex-sums are pairwise distinct, where the vertex-sum of each vertex is the sum of labels over edges incident to this vertex. A graph is antimagic if it has an antimagic labelling. Hartsfield and Ringel in 1990 stated the celebrated conjecture: Every connected graph other than K_2 is antimagic. We prove the conjecture for join graphs with at least three vertices.
Let G = (V(G), E(G)) be a simple undirected graph. A total dominating set S\subseteq V(G) is certified if every vertex in S has either zero or at least two neighbors outside S. The minimum cardinality of such a set is the certified total domination number, denoted by \gamma_{\mathrm{cer}}^{t}(G). We establish its relat...
Jamil J. Hamja, Mohammad Marvic, N. Mohammad et al.· International Journal of Mat...· 0 citations
Let G be a nontrivial graph. A set D ⊆ V(G) is a double dominating set of G if |
\mathrm{N_G}
[v] ∩ D| ≥ 2 for every vertex v ∈ V(G), where
\mathrm{N_G}
[v] represents the closed neighborhood of v. The double domination number of G is the minimum cardinality among all double dominating sets of G. In this paper we...
A. Cabrera-Martínez, Ismael Rios-Villamar, J. Sigarreta· Filomat· 1 citation
How many disjoint copies of some graph $H$ can appear in an induced subgraph of a strongly regular graph? We give upper and lower bounds that depend on the number of vertices in their boundary. This generalizes a result of Lou-Murin which bounds the independence number of a (hitherto still elusive) strongly regular gra...
A graph is locally irregular if adjacent vertices have distinct degrees. A graph G is decomposable if its edge set can be decomposed into locally irregular graphs, and its locally irregular chromatic index lir(G) is the least number of graphs in such a decomposition. We prove that lir(G)<= 18 for every decomposable gra...
C. Lintzmayer, G. Mota, M. Sambinelli et al.· 0 citations
We prove that there is an absolute constant $c>0$ such that every connected vertex-transitive graph $G$ on $n \ge 3$ vertices contains a cycle of length at least $cn$. Moreover, every such graph with sufficiently large degree $d$ contains a cycle of length at least $(1-d^{-1/100})n$. This gives the first linear bound t...
A semitotal dominating set of a nontrivial connected graph G is a dominating set D of G such that every vertex in D is within distance two of another vertex in D. The semitotal domination number of G is the minimum cardinality among all semitotal dominating sets of G. In this article, we investigate the semitotal domin...
A. Cabrera-Martínez, J. L. López-Carmona, José M. Rodríguez et al.· Asian-European Journal of Ma...· 0 citations
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