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Equitable Vertex Arboricity Conjecture Holds for Graphs with Low Degeneracy
The equitable tree-coloring can formulate a structure decomposition problem on the communication network with some security considerations. Namely,...
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Total Equitable List Coloring
An equitable coloring is a proper coloring of a graph such that the sizes of the color classes differ by at most one. A graph G is equitably k -colorab...
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Equitable Coloring of Three Classes of 1-planar Graphs
A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. A plane graph with near-independent...
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Three Early Problems on Size Ramsey Numbers
The size Ramsey number of a graph H is defined as the minimum number of edges in a graph G such that there is a monochromatic copy of H in every...
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A Structure of 1-Planar Graph and Its Applications to Coloring Problems
A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. In this paper, we first give a useful...
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On the Equitable Edge-Coloring of 1-Planar Graphs and Planar Graphs
An edge-coloring of a graph G is equitable if, for each vertex v of G , the number of edges of any one color incident with v differs from the number...
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Embedding Graphs into Larger Graphs: Results, Methods, and Problems
Extremal Graph Theory is a very deep and wide area of modern combinatorics. It is very fast develo**, and in this long but relatively short survey...
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Equitable vertex arboricity of 5-degenerate graphs
Wu et al. (Discret Math 313:2696–2701,
2013 ) conjectured that the vertex set of any simple graph G can be equitably partitioned into m subsets so... -
On Equitable Colorings of Sparse Graphs
A graph is equitably k -colorable if G has a proper vertex k -coloring such that the sizes of any two color classes differ by at most one. Chen, Lih...
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Equitable colorings of Cartesian products of square of cycles and paths with complete bipartite graphs
A graph G is said to be equitably k -colorable if the vertex set of G can be divided into k independent sets for which any two sets differ in size at...
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Equitable Coloring of Graphs
If the vertices of a graph G are colored with k colors such that no adjacent vertices receive the same color and the sizes of any two color classes...
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On the KŁR conjecture in random graphs
The KŁR conjecture of Kohayakawa, Łuczak, and Rödl is a statement that allows one to prove that asymptotically almost surely all subgraphs of the...
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Network Optimization
Many real-life problems can be modeled as optimization problems in networks. Examples include finding shortest paths, scheduling classes in...
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Eigenvalues and Eigenvectors of Graphs
In this chapter, we apply the linear algebra from the previous chapter to graph spectra.
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Proof of the bandwidth conjecture of Bollobás and Komlós
In this paper we prove the following conjecture by Bollobás and Komlós: For every γ > 0 and integers r ≥ 1 and Δ, there exists β > 0 with the...
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Siamese Combinatorial Objects via Computer Algebra Experimentation
Following Kharaghani and Torabi [On a decomposition of complete graphs, Graphs Comb., 19 (2003), 519–526], we introduce new concepts of Siamese color...
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Tiling Transitive Tournaments and Their Blow-ups
Let TT k denote the transitive tournament on k vertices. Let TT ( h , k ) denote the graph obtained from TT k by replacing each vertex with an independent...