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Using General Large Language Models to Classify Mathematical Documents
In this article we report on an initial exploration to assess the viability of using the general large language models (LLMs), recently made public,...
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Rainbow Pancyclicity in a Collection of Graphs Under the Dirac-type Condition
Let G = { G i : i ∈ [ n ]} be a collection of not necessarily distinct n -vertex graphs with the same vertex set V , where G can be seen as an edge-colored...
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Majority choosability of 1-planar digraph
A majority coloring of a digraph D with k colors is an assignment π : V ( D ) → {1, 2, …, k } such that for every v ∈ V ( D ) we have π ( w ) = π ( v ) for at most...
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Rainbow and Monochromatic Vertex-connection of Random Graphs
A vertex-colored path P is rainbow if its internal vertices have distinct colors; whereas P is monochromatic if its internal vertices are colored the...
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Incidence Coloring of Outer-1-planar Graphs
A graph is outer-1-planar if it can be drawn in the plane so that all vertices lie on the outer-face and each edge crosses at most one another edge....
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1-planar graphs with girth at least 6 are (1,1,1,1)-colorable
A graph is 1-planar if it can be drawn in the Euclidean plane so that each edge is crossed by at most one other edge. A 1-planar graph on n vertices...
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Partitioning a planar graph without chordal 5-cycles into two forests
It was known that the vertex set of every planar graph can be partitioned into three forests. We prove that the vertex set of a planar graph without...
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Strong Edge Coloring of Outerplane Graphs with Independent Crossings
The strong chromatic index of a graph is the minimum number of colors needed in a proper edge coloring so that no edge is adjacent to two edges of...
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The Chromatic Number of the Product of 14-Chromatic Graphs Can BE 13
We show that for any n ≥ 13, there exist graphs with chromatic number larger than n whose product has chromatic number n . Our construction is an...
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Acyclic Edge Coloring of 1-planar Graphs without 4-cycles
An acyclic edge coloring of a graph G is a proper edge coloring such that there are no bichromatic cycles in G . The acyclic chromatic index
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Strict Neighbor-Distinguishing Total Index of Graphs
A total-coloring of a graph G is strict neighbor-distinguishing if for any two adjacent vertices u and v , the set of colors used on u and its...
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Maximal Digraphs with Respect to Primitive Positive Constructability
We study the class of all finite directed graphs (digraphs) up to primitive positive constructibility. The resulting order has a unique maximal...
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Every Graph Embedded on the Surface with Euler Characteristic Number ε = −1 is Acyclically 11-choosable
A proper vertex coloring of a graph G is acyclic if there is no bicolored cycles in G . A graph G is acyclically k-choosable if for any list...
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1-planar Graphs without 4-cycles or 5-cycles are 5-colorable
A graph is 1-planar if it can be drawn on the Euclidean plane so that each edge is crossed by at most one other edge. A proper vertex k -coloring of a...
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List 2-distance Coloring of Planar Graphs with Girth Five
A 2-distance coloring of a graph is a coloring of the vertices such that two vertices at distance at most two receive distinct colors. A list...
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2-divisibility of Some Odd Hole Free Graphs
Let G be a graph. We say that G is 2-divisible if for each induced subgraph H of G , either V ( H ) is a stable set, or V ( H ) can be partitioned into two...
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On Decidability of Hyperbolicity
We prove that a wide range of coloring problems for graphs on surfaces can be resolved by inspecting a finite number of configurations.