148 research outputs found
Triangle-free graphs that do not contain an induced subdivision of Kâ‚„ are 3-colorable
We show that triangle-free graphs that do not contain an induced subgraph isomorphic to a subdivision of K4 are 3-colorable. This proves a conjecture of Trotignon and Vušković [J. Graph Theory. 84 (2017), no. 3, pp. 233–248]
On graphs with no induced subdivision of
We prove a decomposition theorem for graphs that do not contain a subdivision
of as an induced subgraph where is the complete graph on four
vertices. We obtain also a structure theorem for the class of graphs
that contain neither a subdivision of nor a wheel as an induced subgraph,
where a wheel is a cycle on at least four vertices together with a vertex that
has at least three neighbors on the cycle. Our structure theorem is used to
prove that every graph in is 3-colorable and entails a polynomial-time
recognition algorithm for membership in . As an intermediate result, we
prove a structure theorem for the graphs whose cycles are all chordless
On wheel-free graphs
A wheel is a graph formed by a chordless cycle and a vertex that has at least three neighbors in the cycle. We prove that every 3-connected graph that does not contain a wheel as a subgraph is in fact minimally 3-connected. We prove that every graph that does not contain a wheel as a subgraph is 3-colorable
A comprehensive introduction to the theory of word-representable graphs
Letters x and y alternate in a word w if after deleting in w all letters but the copies of x and y we either obtain a word xyxy⋯ (of even or odd length) or a word yxyx⋯  (of even or odd length). A graph G=(V,E) is word-representable if and only if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if xy ∈ E.  Word-representable graphs generalize several important classes of graphs such as circle graphs, 3-colorable graphs and comparability graphs. This paper offers a comprehensive introduction to the theory of word-representable graphs including the most recent developments in the area
Turan Problems and Shadows III: expansions of graphs
The expansion of a graph is the -uniform hypergraph obtained
from by enlarging each edge of with a new vertex disjoint from
such that distinct edges are enlarged by distinct vertices. Let
denote the maximum number of edges in a -uniform hypergraph with
vertices not containing any copy of a -uniform hypergraph . The study of
includes some well-researched problems, including the case that
consists of disjoint edges, is a triangle, is a path or cycle,
and is a tree. In this paper we initiate a broader study of the behavior of
. Specifically, we show whenever and . One of the main open problems
is to determine for which graphs the quantity is quadratic in
. We show that this occurs when is any bipartite graph with Tur\'{a}n
number where , and in
particular, this shows where is the
three-dimensional cube graph
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