Injective Chromatic Number Research Articles

Two smaller upper bounds of list injective chromatic number

An injective coloring of a graph $$G$$ is an assignment of colors to the vertices of $$G$$ so that any two vertices with a common neighbor receive distinct colors. Let $$\chi _{i}^{l}(G)$$ denote the list injective chromatic number of $$G$$ . We prove that (1) $$\chi _{i}^{l}(G)=\Delta $$ for a graph $$G$$ with the maximum average degree $$Mad(G)\le \frac{18}{7}$$ and maximum degree $$\Delta \ge 9$$ ; (2) $$\chi _{i}^{l}(G)\le \Delta +2$$ if $$G$$ is a plane graph with $$\Delta \ge 21$$ and without 3-, 4-, 8-cycles.

List injective coloring of planar graphs with girth 5, 6, 8

A vertex coloring of a graph G is called injective if any two vertices with a common neighbor receive distinct colors. Let χi(G), χil(G) be the injective chromatic number and injective choosability number of G, respectively. Suppose that G is a planar graph with maximum degree Δ and girth g. We show that (1) if g≥5 then χil(G)≤Δ+7 for any Δ, and χil(G)≤Δ+4 if Δ≥13; (2) χil(G)≤Δ+2 if g≥6 and Δ≥8; (3) χil(G)≤Δ+1 if g≥8 and Δ≥5.

INJECTIVE COLORING OF PLANAR GRAPHS WITH GIRTH 7

An injective k-coloring of a graph G is an assignment of k colors to the vertices of G so that any two vertices with a common neighbor receive distinct colors, and χi(G) is the injective chromatic number of G. Dimitrov et al. proved χi(G) ≤ Δ(G) + 2 for a planar graph G with g(G) ≥ 7. In this paper, we show that if G is a planar graph with g(G) ≥ 7 and Δ(G) ≥ 7, then χi(G) ≤ Δ(G) + 1.

Some results on the injective chromatic number of graphs

A k-coloring of a graph G=(V,E) is a mapping c:V?{1,2,?,k}. The coloring c is injective if, for every vertex v?V, all the neighbors of v are assigned with distinct colors. The injective chromatic number ? i (G) of G is the smallest k such that G has an injective k-coloring. In this paper, we prove that every K 4-minor free graph G with maximum degree Δ?1 has $\chi_{i}(G)\le \lceil \frac{3}{2}\Delta\rceil$ . Moreover, some related results and open problems are given.

Injective colorings of sparse graphs

Let mad ( G ) denote the maximum average degree (over all subgraphs) of G and let χ i ( G ) denote the injective chromatic number of G . We prove that if mad ( G ) ≤ 5 2 , then χ i ( G ) ≤ Δ ( G ) + 1 ; and if mad ( G ) < 42 19 , then χ i ( G ) = Δ ( G ) . Suppose that G is a planar graph with girth g ( G ) and Δ ( G ) ≥ 4 . We prove that if g ( G ) ≥ 9 , then χ i ( G ) ≤ Δ ( G ) + 1 ; similarly, if g ( G ) ≥ 13 , then χ i ( G ) = Δ ( G ) .

Injective Colorings of Graphs with Low Average Degree

Let mad (G) denote the maximum average degree (over all subgraphs) of G and let χ i (G) denote the injective chromatic number of G. We prove that if Δ≥4 and $\mathrm{mad}(G)<\frac{14}{5}$ , then χ i (G)≤Δ+2. When Δ=3, we show that $\mathrm{mad}(G)<\frac{36}{13}$ implies χ i (G)≤5. In contrast, we give a graph G with Δ=3, $\mathrm{mad}(G)=\frac{36}{13}$ , and χ i (G)=6.

Some bounds on the injective chromatic number of graphs

We give some bounds on the injective chromatic number.

Injective coloring of planar graphs

A coloring of a graph G is injective if its restriction to the neighborhood of any vertex is injective. The injective chromatic number χ i ( G ) of a graph G is the least k such that there is an injective k -coloring. In this paper we prove that if G is a planar graph with girth g and maximum degree Δ , then (1) χ i ( G ) = Δ if either g ≥ 20 and Δ ≥ 3 , or g ≥ 7 and Δ ≥ 71 ; (2) χ i ( G ) ≤ Δ + 1 if g ≥ 11 ; (3) χ i ( G ) ≤ Δ + 2 if g ≥ 8 .

On the injective chromatic number of graphs

We define the concepts of an injective colouring and the injective chromatic number of a graph and give some upper and lower bounds in general, plus some exact values. We explore in particular the injective chromatic number of the hypercube and put it in the context of previous work on similar concepts, especially the theory of error-correcting codes. Finally, we give necessary and sufficient conditions for the injective chromatic number to be equal to the degree for a regular graph.