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Kernels in digraphs with colored vertices

Mucuy-kak Guevara, Teresa I. Hoekstra-Mendoza, Miguel Licona-Velazquez

TL;DR

The paper defines and analyzes up-color kernels in $c$-colored digraphs, establishing when such kernels exist for basic structures (paths, cycles, forests, wheels) and for more complex families (unicyclic and cycles with chords). It then extends the study to graph operations, providing explicit necessary and sufficient criteria for up-color kernels in Cartesian and strong products, Zykov sums, and crown-like constructions, and investigates how these kernels behave under line digraph transformations. A key contribution is the constructive correspondence between up-color kernels of a digraph $D$ and its line digraph $L(D)$ under an outer coloration, together with counterexamples illustrating limits of color-preservation under different colorings. Overall, the work generalizes kernel theory to hierarchically colored digraphs and offers tools to analyze domination-like structures under vertex color orderings in a variety of graph constructions.

Abstract

In this paper, we introduce the concept of up-color kernel, which is a generalization of a kernel for vertex-colored digraphs. We give sufficient and necessary conditions for several families of digraphs to have an up-color kernel, as well as for certain products of digraphs.

Kernels in digraphs with colored vertices

TL;DR

The paper defines and analyzes up-color kernels in -colored digraphs, establishing when such kernels exist for basic structures (paths, cycles, forests, wheels) and for more complex families (unicyclic and cycles with chords). It then extends the study to graph operations, providing explicit necessary and sufficient criteria for up-color kernels in Cartesian and strong products, Zykov sums, and crown-like constructions, and investigates how these kernels behave under line digraph transformations. A key contribution is the constructive correspondence between up-color kernels of a digraph and its line digraph under an outer coloration, together with counterexamples illustrating limits of color-preservation under different colorings. Overall, the work generalizes kernel theory to hierarchically colored digraphs and offers tools to analyze domination-like structures under vertex color orderings in a variety of graph constructions.

Abstract

In this paper, we introduce the concept of up-color kernel, which is a generalization of a kernel for vertex-colored digraphs. We give sufficient and necessary conditions for several families of digraphs to have an up-color kernel, as well as for certain products of digraphs.

Paper Structure

This paper contains 9 sections, 23 theorems, 4 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Up-color kernels in digraphs
  3. Paths, cycles and trees
  4. Wheels, unicyclic graphs and cycles with chords
  5. Products
  6. Cartesian and strong products
  7. Zykov sum
  8. Up-color kernels in $D$, in $D\triangledown \mathcal{H}$ and in $D\vartriangle \mathcal{H}$.
  9. Up-color kernels in $D$ and in $L(D)$

Key Result

Proposition 1

Let $G_{n}$ denote a directed path with $V(G_{n})=\{x_0, \dots, x_{n}\}$, $A(G_{n})=\{(x_i,x_{i-1}):1\leq i\leq n-1\}$ and $c:V(G_{n})\rightarrow \{0,1,\ldots\}$ a $c$-coloring of $G_{n}$. Then $G_{n}$ has an up-color kernel if and only if $c(x_{2i})>c(x_{2i+1})$ for every $i \geq 0.$

Figures (11)

  • Figure 1: $c$-colored path with out an up-color kernel.
  • Figure 2: The black vertices correspond to the even vertices.
  • Figure 3: The purple vertices represent an up-color kernel for $G$ if the conditions of Theorem \ref{['cubo']} hold.
  • Figure 4: The dark purple vertices represent an up-color kernel for $G$ if the conditions of Theorem \ref{['fuerte']} hold.
  • Figure 5: The purple vertices form an up-color kernel for $S_3^+\square S_4^-$ if the conditions of Theorem \ref{['yx']} hold.
  • ...and 6 more figures

Theorems & Definitions (54)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • proof
  • Corollary 1
  • Definition 4
  • Remark 1
  • Theorem 1
  • proof
  • ...and 44 more