相关论文: Odd wheels are not odd-distance graphs

A $hole$ is an induced cycle of length at least four, and an odd hole is a hole of odd length. A {\em fork} is a graph obtained from $K_{1,3}$ by subdividing an edge once. An {\em odd balloon} is a graph obtained from an odd hole by…

A k-outerplanar graph is a graph that can be drawn in the plane without crossing such that after k-fold removal of the vertices on the outer-face there are no vertices left. In this paper, we study how to triangulate a k-outerplanar graph…

A graph admitting an automorphism with two orbits of the same length is called a bicirculant. Recently, Jajcay et al. initiated the investigation of the edge-transitive bicirculants with the properties that one of the subgraphs induced by…

In an undirected graph, the odd cycle packing number is the maximum number of pairwise vertex-disjoint odd cycles. The odd cycle transversal number is the minimum number of vertices that hit every odd cycle. The maximum ratio between…

The odd-ballooning of a graph $G$, denoted by $G_q$, is the graph obtained from replacing each edge in $G$ by a odd cycle of the same size where the new vertices of the odd cycles are all different. In 2002, Erd\"os et al. determined the…

A circle graph is an intersection graph of a set of chords of a circle. We describe the unavoidable induced subgraphs of circle graphs with large treewidth. This includes examples that are far from the `usual suspects'. Our results imply…

The main result of this article is the decomposition of tensor products of representations of SL(2) in the sum of irreducible representations parametrized by outerplanar graphs. An outerplanar graph is a graph with the vertices 0, 1, 2,…

Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. In this paper we provide several novel characterizations of planar median…

Acyclic and cyclic orientations of an undirected graph have been widely studied for their importance: an orientation is acyclic if it assigns a direction to each edge so as to obtain a directed acyclic graph (DAG) with the same vertex set;…

A pseudoline arrangement graph is a planar graph induced by an embedding of a (simple) pseudoline arrangement. We study the corresponding graph realization problem and properties of pseudoline arrangement graphs. In the first part, we give…

Odd coloring is a proper coloring with an additional restriction that every non-isolated vertex has some color that appears an odd number of times in its neighborhood. The minimum number of colors $k$ that can ensure an odd coloring of a…

The transmission of a vertex $v$ of a (chemical) graph $G$ is the sum of distances from $v$ to other vertices in $G$. If any two vertices of $G$ have different transmissions, then $G$ is a transmission irregular graph. It is shown that for…

A graph is called odd (respectively, even) if every vertex has odd (respectively, even) degree. Gallai proved that every graph can be partitioned into two even induced subgraphs, or into an odd and an even induced subgraph. We refer to a…

The vertex-edge incidence matrix of a (connected) unicyclic graph G is a square matrix which is invertible if and only if the cycle of G is an odd cycle. A combinatorial formula of the inverse of the incidence matrix of an odd unicyclic…

An extremal graph for a graph $H$ on $n$ vertices is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $T_{n,r}$ be the Tur\'{a}n graph, which is the complete $r$-partite graph on $n$ vertices…

A pseudoline is a homeomorphic image of the real line in the plane so that its complement is disconnected. An arrangement of pseudolines is a set of pseudolines in which every two cross exactly once. A drawing of a graph is pseudolinear if…

A biased graph is a graph with a class of selected circles ("cycles", "circuits"), called balanced, such that no theta subgraph contains exactly two balanced circles. A biased graph $\Omega$ has two natural matroids, the frame matroid…

We prove that every graph $G$ on $n$ vertices with no isolated vertices contains an induced subgraph of size at least $n/10000$ with all degrees odd. This solves an old and well-known conjecture in graph theory.

We study the reconfiguration of odd matchings of combinatorial graphs. Odd matchings are matchings that cover all but one vertex of a graph. A reconfiguration step, or flip, is an operation that matches the isolated vertex and,…

A graph is called odd if there is an orientation of its edges and an automorphism that reverses the sense of an odd number of its edges, and even otherwise. Pontus von Br\"omssen (n\'e Andersson) showed that the existence of such an…