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We confirm the equitable $Delta$-coloring conjecture for interval graphs and establish the monotonicity of equitable colorability for them. We further obtain results on equitable colorability about square (or Cartesian) and cross (or direct) products of graphs.
If the vertices of a graph $G$ are colored with $k$ colors such that no adjacent vertices receive the same color and the sizes of any two color classes differ by at most one, then $G$ is said to be equitably $k$-colorable. Let $|G|$ denote the number
An equitable $k$-partition of a graph $G$ is a collection of induced subgraphs $(G[V_1],G[V_2],ldots,G[V_k])$ of $G$ such that $(V_1,V_2,ldots,V_k)$ is a partition of $V(G)$ and $-1le |V_i|-|V_j|le 1$ for all $1le i<jle k$. We prove that every planar
Let $G = (V, E)$ be a finite simple undirected graph without $K_2$ components. A bijection $f : E rightarrow {1, 2,cdots, |E|}$ is called a {bf local antimagic labeling} if for any two adjacent vertices $u$ and $v$, they have different vertex sums, i
For a given graph $G$, the least integer $kgeq 2$ such that for every Abelian group $mathcal{G}$ of order $k$ there exists a proper edge labeling $f:E(G)rightarrow mathcal{G}$ so that $sum_{xin N(u)}f(xu) eq sum_{xin N(v)}f(xv)$ for each edge $uvin E
The textit{$k$-weak-dynamic number} of a graph $G$ is the smallest number of colors we need to color the vertices of $G$ in such a way that each vertex $v$ of degree $d(v)$ sees at least $rm{min}{k,d(v)}$ colors on its neighborhood. We use reducible