The majority coloring of the join and Cartesian product of some digraph

A majority coloring of a digraph is a vertex coloring such that for every vertex, the number of vertices with the same color in the out-neighborhood does not exceed half of its out-degree. Kreutzer, Oum, Seymour and van der Zyper proved that every digraph is majority 4-colorable and conjecture that every digraph has a majority 3-coloring. This paper mainly studies the majority coloring of the joint and Cartesian product of some special digraphs and proved the conjecture is true for the join graph and the Cartesian product. According to the influence of the number of vertices in digraph, we prove the majority coloring of the joint and Cartesian product of some digraph.

Computing the Atom Graph of a Graph and the Union Join Graph of a Hypergraph

The atom graph of a graph is a graph whose vertices are the atoms obtained by clique minimal separator decomposition of this graph, and whose edges are the edges of all possible atom trees of this graph. We provide two efficient algorithms for computing this atom graph, with a complexity in O(min(nωlogn,nm,n(n+m¯)) time, where n is the number of vertices of G, m is the number of its edges, m¯ is the number of edges of the complement of G, and ω, also denoted by α in the literature, is a real number, such that O(nω) is the best known time complexity for matrix multiplication, whose current value is 2,3728596. This time complexity is no more than the time complexity of computing the atoms in the general case. We extend our results to α-acyclic hypergraphs, which are hypergraphs having at least one join tree, a join tree of an hypergraph being defined by its hyperedges in the same way as an atom tree of a graph is defined by its atoms. We introduce the notion of union join graph, which is the union of all possible join trees; we apply our algorithms for atom graphs to efficiently compute union join graphs.

The Local Antimagic Chromatic Numbers of Some Join Graphs

Let G=(V(G),E(G)) be a connected graph with n vertices and m edges. A bijection f:E(G)→{1,2,⋯,m} is an edge labeling of G. For any vertex x of G, we define ω(x)=∑e∈E(x)f(e) as the vertex label or weight of x, where E(x) is the set of edges incident to x, and f is called a local antimagic labeling of G, if ω(u)≠ω(v) for any two adjacent vertices u,v∈V(G). It is clear that any local antimagic labelling of G induces a proper vertex coloring of G by assigning the vertex label ω(x) to any vertex x of G. The local antimagic chromatic number of G, denoted by χla(G), is the minimum number of different vertex labels taken over all colorings induced by local antimagic labelings of G. In this paper, we present explicit local antimagic chromatic numbers of Fn∨K2¯ and Fn−v, where Fn is the friendship graph with n triangles and v is any vertex of Fn. Moreover, we explicitly construct an infinite class of connected graphs G such that χla(G)=χla(G∨K2¯), where G∨K2¯ is the join graph of G and the complement graph of complete graph K2. This fact leads to a counterexample to a theorem of Arumugam et al. in 2017, and our result also provides a partial solution to Problem 3.19 in Lau et al. in 2021.

FINITE GROUPS WITH THE SAME JOIN GRAPH AS A FINITE NILPOTENT GROUP

Given a finite group G, we denote by Δ(G) the graph whose vertices are the proper subgroups of G and in which two vertices H and K are joined by an edge if and only if G = ⟨H, K⟩. We prove that if there exists a finite nilpotent group X with Δ(G) ≅ Δ(X), then G is supersoluble.

A Note on Edge Irregularity Strength of Some Graphs

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>Let </span><em>G</em><span>(</span><span><em>V</em>, <em>E</em></span><span>) </span><span>be a finite simple graph and </span><span>k </span><span>be some positive integer. A vertex </span><em>k</em><span>-labeling of graph </span><em>G</em>(<em>V,E</em>), Φ : <em>V</em> → {1,2,..., <em>k</em>}, is called edge irregular <em>k</em>-labeling if the edge weights of any two different edges in <em>G</em> are distinct, where the edge weight of <em>e</em> = <em>xy</em> ∈ <em>E</em>(<em>G</em>), w_Φ(e), is defined as <em>w</em>_Φ(<em>e</em>) = Φ(<em>x</em>) + Φ(<em>y</em>). The edge irregularity strength for graph G is the minimum value of k such that Φ is irregular edge <em>k</em>-labeling for <em>G</em>. In this note we derive the edge irregularity strength of chain graphs <em>mK</em>₃−path for m ≢ 3 (mod4) and <em>C</em>[<em>C_n</em>^(<em>m</em>)] for all positive integers <em>n</em> ≡ 0 (mod 4) 3<em>n</em> and <em>m</em>. We also propose bounds for the edge irregularity strength of join graph <em>P_m</em> + <em>Ǩ_n</em> for all integers <em>m, n</em> ≥ 3.</p></div></div></div>

Resistance Distances and Kirchhoff Index in Generalised Join Graphs

The resistance distance between any two vertices of a connected graph is defined as the effective resistance between them in the electrical network constructed from the graph by replacing each edge with a unit resistor. The Kirchhoff index of a graph is defined as the sum of all the resistance distances between any pair of vertices of the graph. Let G=H[G1, G2, …, Gk ] be the generalised join graph of G1, G2, …, Gk determined by H. In this paper, we first give formulae for resistance distances and Kirchhoff index of G in terms of parameters of ${G'_i}s$ and H. Then, we show that computing resistance distances and Kirchhoff index of G can be decomposed into simpler ones. Finally, we obtain explicit formulae for resistance distances and Kirchhoff index of G when ${G'_i}s$ and H take some special graphs, such as the complete graph, the path, and the cycle.