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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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Edge-Partitioning a Graph into Paths: Beyond the Barát-Thomassen Conjecture
In 2006, Barát and Thomassen conjectured that there is a function f such that, for every fixed tree T with t edges, every f ( t )-edge-connected graph...
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Some class 1 graphs on g c -colorings
An edge-coloring of a graph G is an assignment of colors to all the edges of G . A g c -coloring of a graph G is an edge-coloring of G such that each...
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Leader selection for strong structural controllability of single-integrator multi-agent systems
This paper addresses the leader selection problem for strong structural controllability (SSC) of multi-agent systems (MASs). For a path-bud graph, it...
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Some Hard Stable Marriage Problems: A Survey on Multivariate Analysis
We survey an emerging area of research within algorithmic game theory: multivariate analysis of games. This article surveys the landscape of work on...
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Degree Conditions
In this chapter, we consider results which use assumptions on the degrees or number of edges, thereby driving the proper connection number down.
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An exact approach for the balanced k-way partitioning problem with weight constraints and its application to sports team realignment
In this work a balanced k -way partitioning problem with weight constraints is defined to model the sports team realignment. Sports teams must be...
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The Ramsey Numbers for A Triple of Long Cycles
We find the asymptotic value of the Ramsey number for a triple of long cycles, where the lengths of the cycles are large but may have different...
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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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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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Hypergraph cuts above the average
An r -cut of a k -uniform hypergraph H is a partition of the vertex set of H into r parts and the size of the cut is the number of edges which have a...
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A tight lower bound for Szemerédi’s regularity lemma
We determine the order of the tower height for the partition size in a version of Szemerédi’s regularity lemma. This addresses a question of Gowers.
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Perfect 2-colorings of the generalized Petersen graph
In this paper, we enumerate the parameter matrices of all perfect 2-colorings of the generalized Petersen graphs GP ( n ,2), where n ≥ 5. We also...
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Decomposition methods based on articulation vertices for degree-dependent spanning tree problems
Decomposition methods for optimal spanning trees on graphs are explored in this work. The attention is focused on optimization problems where the...
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Degree Conditions for the Existence of Vertex-Disjoint Cycles and Paths: A Survey
In this paper, we survey results and conjectures on degree conditions for the existence of vertex-disjoint cycles and paths. In particular, we focus...
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A refinement of a result of Corrádi and Hajnal
Corrádi and Hajnal proved that for every k ≥ 1 and n ≥ 3 k , every n -vertex graph with minimum degree at least 2 k contains k vertex-disjoint cycles....