What question did this study set out to answer?

The study aims to establish bounds on the degrees of vertices in graphs, facilitating the existence of specific vertex colorings.

Distinguishing finite and infinite trees of arbitrary cardinality

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This analysis reveals limits on vertex colorings in trees, implying important distinctions for graph structures.

The study aims to establish bounds on the degrees of vertices in graphs, facilitating the existence of specific vertex colorings.
Investigated automorphisms in finite and infinite graphs
Derived bounds for vertex colorings based on vertex degrees
Analyzed properties of trees with finite and infinite cardinality
For finite trees, established that vertex degree must satisfy $elta(T)\leq2^{m(T)/2}$ for valid colorings
Found that the number of mutually inequivalent colorings for infinite trees is $2^{|T|}$
Proved that for tree-like graphs with degree $Delta(G)\le 2^{aleph_0}$, the number of colorings is also $2^{|G|}$

Imrich et al. (2026) studied this question.

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View Full Paper

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Authors

This analysis reveals limits on vertex colorings in trees, implying important distinctions for graph structures.

The study aims to establish bounds on the degrees of vertices in graphs, facilitating the existence of specific vertex colorings.
Investigated automorphisms in finite and infinite graphs
Derived bounds for vertex colorings based on vertex degrees
Analyzed properties of trees with finite and infinite cardinality
For finite trees, established that vertex degree must satisfy $elta(T)\leq2^{m(T)/2}$ for valid colorings
Found that the number of mutually inequivalent colorings for infinite trees is $2^{|T|}$
Proved that for tree-like graphs with degree $Delta(G)\le 2^{aleph_0}$, the number of colorings is also $2^{|G|}$

Imrich et al. (2026) studied this question.

View Full Paper

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