WebMar 1, 2024 · Note that such an edge-coloring is not necessarily proper. The minimum number of colors required for an injective edge-coloring is called the injective chromatic index of G, denoted by χ i ′ ( G ). For every integer k ≥ 2, we show that every k-degenerate graph G with maximum degree Δ satisfies χ i ′ ( G ) ≤ ( 4 k − 3 ) Δ − 2 k 2 − k + 3.
Complexity and algorithms for injective edge-coloring in graphs
WebAn injective k-edge-coloring of a graph G is an assignment of colors, i.e. integers in f1;:::;kg, to ... Moreover all subcubic planar bipartite graphs are injectively 4-edge-colorable [14]. Note that in [1], this notion is studied as the inducde star arboricity of a graph, that is, the smallest WebA vi-simultaneous proper k-coloring of a graph G is a coloring of all vertices and incidences of the graph in which any two adjacent or incident elements in the set V(G)∪I(G) receive distinct colors, where I(G) is the set of incidences of G.The vi-simultaneous chromatic number, denoted by χ vi (G), is the smallest integer k such that G has a vi-simultaneous … hillard box ins tyler
20a-tree-cotree-structures
WebAbstract A k -injective edge coloring of a graph G is a coloring f: E ( G) → C = { 1, 2, 3, …, k }, such that if e 1, e 2 and e 3 are consecutive edges in G, then f ( e 1) ≠ f ( e 3). χ i ′ ( G) = … WebOct 1, 2024 · Injective edge-coloring of graphs with given maximum degree Alexandr Kostochka, André Raspaud, Jingwei Xu A coloring of edges of a graph G is injective if for any two distinct edges e_1 and e_2, the colors of e_1 and e_2 are distinct if they are at distance 1 in G or in a common triangle. WebJul 23, 2024 · An injective edge-coloring of a graph is an edge-coloring such that if , , and are three consecutive edges in (they are consecutive if they form a path or a cycle of length three), then and receive different colors. The minimum integer such that, has an injective edge-coloring with colors, is called the injective chromatic index of ( ). hillard clutch manual