Universal families of rayless graphs

TL;DR

The paper resolves the strong universality landscape for rayless graphs across all infinite cardinals. By a rank-based inductive construction, it proves that the strong complexity of the class of $κ$-rayless graphs, $StCpx(A_κ)$, equals $κ^+$, and it provides explicit strongly universal families of that size. It then analyzes how forbidding finite or infinite subgraphs alters this complexity, identifying cases where it remains $κ^+$ and cases where it escalates to the continuum, notably via a countable rank-1 graph that forces maximal complexity. The work further shows that several natural rayless subclasses (trees, bipartite rayless graphs, graphs without even cycles, and graphs without infinite trails) preserve the minimal strong complexity, offering a comprehensive characterization of universality under various restrictions.

Abstract

We study the existence and cardinality of universal families for classes of rayless graphs. It is known, by a result of Diestel, Halin, and Vogler, that the class of countable rayless graphs does not admit a countable universal family, leaving open the precise complexity of this class. We prove that for every infinite cardinal $κ$, the class of rayless graphs of cardinality at most $κ$ admits a strongly universal family of size exactly $κ^+$, and that no smaller family can exist. This settles the problem for the countable case and extends uniformly to higher cardinalities. We further investigate subclasses defined by forbidding subgraphs. When finitely many finite graphs are forbidden, the strong complexity remains $κ^+$, except in degenerate cases where it collapses to countable. In contrast, the class of countable rayless graphs when forbidding certain infinite graphs has a complexity that reaches its maximum possible value, the continuum. Finally, we establish that natural subclasses -- including rayless trees, bipartite rayless graphs, graphs without even cycles, and graphs without infinite trails -- retain the minimal strong complexity $κ^+$. These results provide a comprehensive characterization of universality in rayless graphs and highlight both its stability under restrictions and its sensitivity to specific obstructions.

Figures (5)

• Figure 1: The construction
• Figure 2: The Graph $K$
• Figure 3: $G_X(7)$ for $X=\{4,5,7,9,...\}$
• Figure 4: Example for $X=\{4,5,7,9,...\}$
• Figure 5: An example of a bouquet.

Theorems & Definitions (47)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5: Schm
• Definition 2.6
• Remark 2.7
• Proposition 2.8
• Theorem 3.1
• Proposition 3.2