Long-eared digraphs

TL;DR

The paper investigates long-ear digraphs by introducing the LE_i families through ear decompositions and exploring kernels, quasi-kernels, and colorings. It develops kernel-transfer lemmas along ears, proves strong structural results such as $\chi(D)\le 3$ for LE_2 and $\chi_o(D)\le 6$ for LE_3, and establishes that Seymour's Second Neighborhood Conjecture and related independent-longest-path properties hold in LE_2, while small quasi-kernels are guaranteed in LE_3. It also demonstrates unbounded oriented chromatic number for LE_2 and provides tightness results for LE_3, highlighting the nuanced impact of ear length on coloring and kernel properties. Collectively, the work deepens understanding of how long-ear structure constrains kernel existence and coloring complexity in digraphs.

Abstract

Let $H$ be a subdigraph of a digraph $D$. An ear of $H$ in $D$ is a path or a cycle in $D$ whose ends lie in $H$ but whose internal vertices do not. An \emph{ear decomposition} of a strong digraph $D$ is a nested sequence $(D_0,D_1,\ldots , D_k)$ of strong subdigraphs of $D$ such that: 1) $D_0$ is a cycle, 2) $D_{i+1} = D_i\cup P_i$, where $P_i$ is an ear of $D_i$ in $D$, for every $i\in \{0,1,\ldots,k-1\}$, and 3) $D_k=D$. In this work, the $\mathcal{LE}_i$ is defined as the family of strong digraphs, with an ear decomposition such that every ear has a length of at least $i\geq 1$. It is proved that Seymour's second Neighborhood Conjecture and the Laborde, Payan, and Soung conjecture, are true in the family $\mathcal{LE}_2$, and the Small quasi-kernel conjecture is true for digraphs in $\mathcal{LE}_3$. Also, some sufficient conditions for a strong nonseparable digraph in $\mathcal{LE}_2$ with a kernel to imply that the previous (following) subdigraph in the ear decomposition has a kernel too, are presented. It is proved that digraphs in $\mathcal{LE}_2$ have a chromatic number at most 3, and a dichromatic number 2 or 3. Finally, the oriented chromatic number of asymmetrical digraphs in $\mathcal{LE}_3$ is bounded by 6, and it is shown that the oriented chromatic number of asymmetrical digraphs in $\mathcal{LE}_2$ is not bounded.

Figures (5)

• Figure 1: Nonplanar digraph in $\mathcal{LE}_2$
• Figure 2: Digraph in $\mathcal{LE}_2$
• Figure 3: Tournament $T_{48}$ with 6 vertices
• Figure 4: Digraph $D$ with an ear decomposition such that every ear has a length 3, and $\chi_o(D)=6$
• Figure 5: Digraphs with $G_1$ and $G_2$

Theorems & Definitions (28)

• Conjecture 1
• Theorem 2
• Conjecture 3: LabordePayanXuong82
• Theorem 4
• proof
• Conjecture 5
• Theorem 6
• proof
• Lemma 7
• proof