WebA 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. Naturally, the injective chromatic index of G, χ inj (G), is the minimum number of colors needed for an injective edge-coloring of G.
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WebEquivalently, suppose we color the edges of \(\Sigma\) red, green, or blue in arbitrary order so that (1) every cycle contains at least one blue edge, (2) every edge cut contains at least one red edge, and (3) an edge is green if and only if it cannot be colored either red or blue. Then the red, green, and blue edges respectively define the ... WebFeb 2, 2024 · An injective edge coloring of a graph G = (V, E) is a coloring c of the edges of G such that if e1,e2 and e3 are consecutive edges in G, then c (e1) c (e3). The injective …
WebJan 7, 2024 · Note that an injective edge coloring is not necessarily a proper edge coloring. The notion of injective edge coloring was introduced in 2015 by Cardoso et al. ( 2024) … WebThe injective edge coloring number or the injective edge chromatic index of a graph G, χ′ i (G), is the minimum number of colors permitted in an i-edge coloring. In the same paper, they gave the exact values of the injective edge coloring number for several classes of graphs, such as path, complete bipartite graph, complete graph and so on.
WebOct 27, 2024 · Injective chromatic index is closely related to strong edge-coloring. A proper injective edge-coloring is exactly a strong edge-coloring, which partitions the edges of a … WebMay 19, 2024 · In this paper, we consider the injective edge coloring numbers of generalized Petersen graphs P ( n, 1) and P ( n, 2). We determine the exact values of injective edge coloring numbers for P ( n, 1) with n ≥ 3, and for P ( n, 2) with 4 ≤ n ≤ 7. For n ≥ 8, we show that 4 ≤ χ i ′ ( P ( n, 2)) ≤ 5. Keywords: k -injective edge coloring,
WebOct 4, 2024 · A k -injective edge-coloring of a graph G is an edge-coloring of G , (not necessarily proper), such that if edges e 1 , e 2 , e 3 are consecutive, then e 1 and e 3 receive distinct colors....
WebAn injective edge coloring of a graph G = (V, E) is a coloring c of the edges of G such that if e 1, e 2 and e 3 are consecutive edges in G, then c (e 1) ≠ c (e 3 ). The injective edge coloring number χ i (G) is the minimum number of colors permitted in such a coloring. chirbes crematorioWebAn injective k -edge coloring of a graph G = ( V ( G), E ( G)) is a k -edge coloring φ such that if e 1 and e 2 are at distance exactly 2 or in the same triangle, then φ ( e 1) ≠ φ ( e 2). The injective chromatic index of G, denoted by χ i ′ ( G), is the minimum k such that G has an injective k -edge coloring. The edge weight of G is ... chirbit childWebAbstract 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) = … graphic designers famous womenWebNote on injective edge-coloring of graphs @article{Miao2024NoteOI, title={Note on injective edge-coloring of graphs}, author={Zhengke Miao and Yimin Song and Gexin Yu}, journal={Discret. Appl. graphic designer self promotion examplesWebJul 15, 2024 · An injective coloring of a graph is a vertex coloring such that any pair of vertices having a common neighbor receives distinct colors. Panda and Priyamvada [ 12 ] proved complexity results for some subclasses of bipartite graphs, which can be re-interpreted as results on exact square coloring. graphic designer shiseido nyc salaryWebOct 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. chirbit daddyWebA 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 … graphic designer sharper image michigan