In this paper, we prove that a cubic line graph G on n vertices rather than the complete graph K4 has b3c vertex-disjoint triangles and the vertex independence number b3c. Moreover, the equitable chromatic number, acyclic chromatic number and bipartite density of G are 3, 3, 79 respectively.

We study the Equitable Connected Partition problem, which is the problem of partitioning a graph into a given number of partitions, such that each partition induces a connected subgraph, and the partitions differ in size by at most one. We examine the problem from the parameterized complexity perspective with respect to the number of partitions, the treewidth, the pathwidth, the size of a minim...

For each positive integer \(k\), a simple graph \(G\) of order \(p\) is said to be \(k\)-prime labeling if there exists an injective function \(f\) whose labels are from \(k\) \(k+p-1\) that induces \(f^{+}:E(G)\to N\) the edges defined by \(f^{+}(uv)=\gcd(f(u),f(v))\), \(\forall\) \(e=uv \in E(G)\) such every pair neighbouring vertices relatively prime. This type known as graph. In this paper,...

A graph $G$ is $k$-$weighted-list-antimagic$ if for any vertex weighting $\omega\colon V(G)\to\mathbb{R}$ and list assignment $L\colon E(G)\to2^{\mathbb{R}}$ with $|L(e)|\geq |E(G)|+k$ there exists an edge labeling $f$ such that $f(e)\in L(e)$ all $e\in E(G)$, labels of edges are pairwise distinct, the sum on incident to a plus weight distinct from at every other vertex. In this paper we prove ...

The equitable tree-coloring can formulate a structure decomposition problem on the communication network with some security considerations. Namely, an tree-k-coloring of graph is vertex coloring using k distinct colors such that every color class induces forest and sizes any two classes differ by at most one. In this paper, we show theoretical results graphs proving d-degenerate maximum degree ...

The paper is devoted to the combinatorial problem concerning equitable colorings of non-uniform simple hypergraphs. Let H = (V,E) be a hypergraph, a coloring with r colors of its vertex set V is called equitable if it is proper (i.e. none of the edges is monochromatic) and the cardinalities of the color classes differ by at most one. We show that if H is a simple hypergraph with minimum edge-ca...