Related papers: Signed-eliminable graphs and free multiplicities o…
We obtain several new upper bounds of the odd graceful chromatic number of a graph $G$, which must be bipartite. Some of our bounds depend only on the number of the vertices of $G$ or the chromatic number of some graphs related to the…
Jerrum, Sinclair and Vigoda (2004) showed that the permanent of any square matrix can be estimated in polynomial time. This computation can be viewed as approximating the partition function of edge-weighted matchings in a bipartite graph.…
Motivated by very large-scale communication networks, we newly introduce exponentiation of graphs. Using the exponential operation on graphs, we can construct various graphs of multi-exponential order with logarithmic diameter. We show that…
The graph braid group of a complete bipartite graph is the fundamental group of a configuration space of points on the graph, which is a CAT(0) cube complex. We combine an analysis of the topology of links of vertices in this complex, the…
A $(a,b)$-coloring of a graph $G$ associates to each vertex a $b$-subset of a set of $a$ colors in such a way that the color-sets of adjacent vertices are disjoint. We define general reduction tools for $(a,b)$-coloring of graphs for $2\le…
A majority edge-coloring of a graph without pendant edges is a coloring of its edges such that, for every vertex $v$ and every color $\alpha$, there are at most as many edges incident to $v$ colored with $\alpha$ as with all other colors.…
The degree sequence of a graph is a numerical method to characterize the properties of graphs. Generalized forms of degree sequences exist for complete graphs and complete graphs. Nikolopolus et al. characterized the number of spanning…
We define the independence ratio and the chromatic number for bounded, self-adjoint operators on an L^2-space by extending the definitions for the adjacency matrix of finite graphs. In analogy to the Hoffman bounds for finite graphs, we…
Inspired by notorious combinatorial optimization problems on graphs, in this paper we consider a series of related problems defined using a metric space and topology determined by a graph. Particularly, we present the Independent Set,…
We consider edge colorings of a graph in such a way that each two different triangles have distinct colorings. It is an extension of the well-known idea of distinguishing all maximal stars in a graph. It was introduced in literature in 1985…
This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and…
Unlabeled multigraphs have diverse applications across scientific fields, from transportation and social networks to polymer physics. In particular, multigraphs are essential for studying the relationship between the spatial organization…
Let $G$ be a simple graph on the vertex set $\{v_1,\dots,v_n\}$ with edge set $E$. Let $K$ be a field. The graphical arrangement $\mathcal{A}_G$ in $K^n$ is the arrangement $x_i-x_j=0, v_iv_j \in E$. An arrangement $\mathcal{A}$ is…
For signed graphs we provide a cubic polynomial upper bound on the multiplicity of its eigenvalues. We show that this bound is sharp by providing examples of signed graphs in which it is attained. We also discuss particular cases in which…
Ideal subarrangements of a Weyl arrangement are proved to be free by the multiple addition theorem (MAT) due to Abe-Barakat-Cuntz-Hoge-Terao (2016). They form a significant class among Weyl subarrangements that are known to be free so far.…
Binary classification problems can be naturally modeled as bipartite graphs, where we attempt to classify right nodes based on their left adjacencies. We consider the case of labeled bipartite graphs in which some labels and edges are not…
We consider the braid groups $\mathbf{B}_n(X)$ on finite simplicial complexes $X$, which are generalizations of those on both manifolds and graphs that have been studied already by many authors. We figure out the relationships between…
We study the following question: how few edges can we delete from any $H$-free graph on $n$ vertices in order to make the resulting graph $k$-colorable? It turns out that various classical problems in extremal graph theory are special cases…
A bipartite graph is chordal bipartite if every cycle of length at least 6 has a chord in it. In this paper, we investigate the structure of $P_5$-free chordal bipartite graphs and show that these graphs have a Nested Neighborhood Ordering,…
Finite type invariants (also known as Vassiliev invariants) of pure braids are considered from a group-theoretic point of view. New results include a construction of a universal invariant with integer coefficients based on the Magnus…