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A graph G is k-edge-choosable if every instance of list edge-coloring that has G as its underlying graph and that provides at least k allowed colors for each edge of G has a proper coloring. The edge choosability, or list edge colorability, list edge chromatic number, or list chromatic index, ch'(G) of graph G is the least number k such that G ...
Note that if c is a proper (Δ+1)-edge-coloring of G then every vertex has a missing color with respect to c. Suppose that no proper (Δ+1)-edge-coloring of G exists. This is equivalent to this statement: (1) Let xy ∈ E and c be arbitrary proper (Δ+1)-edge-coloring of G − xy and α be missing from x and β be missing from y with respect to c.
It is straightforward to test whether a graph may be edge colored with one or two colors, so the first nontrivial case of edge coloring is testing whether a graph has a 3-edge-coloring. As Kowalik (2009) showed, it is possible to test whether a graph has a 3-edge-coloring in time O(1.344 n), while using only polynomial space. Although this time ...
If G is a graph, the line graph L(G) has a vertex for each edge of G, and an edge for each pair of adjacent edges in G. Thus, the chromatic number of L(G) equals the chromatic index of G. If G is bipartite, the cliques in L(G) are exactly the sets of edges in G sharing a common endpoint. Now Kőnig's line coloring theorem, stating that the ...
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The width of the patch also influences diversity: an edge patch must be more pronounced than just a stark border in order to develop gradients of edge effects. Animals traveling between communities can create travel lanes along borders, which in turn increases light reaching plants along the lanes and promotes primary production. As more light ...
The first is an edge effect. [13] This effect originates from the ignorance of interdependences that occur outside the bounded region. [13] Griffith [14] [8] and Griffith and Amrhein [15] highlighted problems according to the edge effect. A typical example is a cross-boundary influence such as cross-border jobs, services and other resources ...
About 150 of the 500 cameras are out of commission due to "several technical problems," NBC News reported, citing an internal Border Patrol memo.