(2,k)-Factor-Critical Graphs and Toughness by Cai M.-C., Favaron O., Li H.

By Cai M.-C., Favaron O., Li H.

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31 35 38 41 45 46 INTRODUCTION Although the first mention of a graph was not until 1878, graph-theoretical ideas can be traced back to 1735 when Leonhard Euler (1707–83) presented his solution of the K¨onigsberg bridges problem. This chapter summarizes some important strands in the development of graph theory since that time. Further information can be found in [BiLlWi98] or [Wi99]. 1 Traversability The origins of graph theory can be traced back to Euler’s work on the K¨onigsberg bridges problem (1735), which subsequently led to the concept of an eulerian graph.

31 35 38 41 45 46 INTRODUCTION Although the first mention of a graph was not until 1878, graph-theoretical ideas can be traced back to 1735 when Leonhard Euler (1707–83) presented his solution of the K¨onigsberg bridges problem. This chapter summarizes some important strands in the development of graph theory since that time. Further information can be found in [BiLlWi98] or [Wi99]. 1 Traversability The origins of graph theory can be traced back to Euler’s work on the K¨onigsberg bridges problem (1735), which subsequently led to the concept of an eulerian graph.

1 Building Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Integer-Valued Invariants . . . . . . . . . . . . . . . . . . . . . 4 Criterion Qualification . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 22 25 28 30 INTRODUCTION Whenever a property of graphs is defined, a family of graphs — those with that property — results.

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