Let P (G,λ) be the chromatic polynomial of a graph G. Two graphs G and H are said to be chromatically equivalent, denoted G ∼ H, if P (G,λ) = P (H,λ). We write [G] = {H |H ∼ G}. If [G] = {G}, then G is said to be chromatically unique. A K4-homeomorph denoted by K4(a, b, c, d, e, f) if the six edges of complete graph K4 are replaced by the six paths of length a, b, c, d, e, f respectively. In th...

Let $G$ be a non-abelian finite group. In this paper, we prove that $Gamma(G)$ is $K_4$-free if and only if $G cong A times P$, where $A$ is an abelian group, $P$ is a $2$-group and $G/Z(G) cong mathbb{ Z}_2 times mathbb{Z}_2$. Also, we show that $Gamma(G)$ is $K_{1,3}$-free if and only if $G cong {mathbb{S}}_3,~D_8$ or $Q_8$.

For each $$n\ge 14$$ , we provide an example of a linklessly embeddable, Tutte-4-connected graph order n. We start with fourteen, and perform 4-vertex splittings to inductively build the family triangle free, 4-connected graphs. prove graphs constructed are as minors clique sums over $$K_4$$ embeddable

let $g$ be a non-abelian finite group. in this paper, we prove that $gamma(g)$ is $k_4$-free if and only if $g cong a times p$, where $a$ is an abelian group, $p$ is a $2$-group and $g/z(g) cong mathbb{ z}_2 times mathbb{z}_2$. also, we show that $gamma(g)$ is $k_{1,3}$-free if and only if $g cong {mathbb{s}}_3,~d_8$ or $q_8$.