A k-total-coloring of a graph G is a coloring of vertex set and edge set using k colors such that no two adjacent or incident elements receive the same color. In this paper, we prove that if G is a planar graph with maximum ∆ ≥ 8 and every 6-cycle of G contains at most one chord or any chordal 6-cycles are not adjacent, then G has a (∆ + 1)-total-coloring.

If G is a simple graph with minimum degree <5(G) satisfying <5(G) ^ f(| K(C?)| -f-1) the total chromatic number conjecture holds; moreover if S(G) ^ f| V(G)\ then #T(G) < A(G) + 3. Also if G has odd order and is regular with d{G) ^ \^/1\V{G)\ then a necessary and sufficient condition for ^T((7) = A((7)+ 1 is given.

A subset D of vertices of a graph G is a dominating set if for each u ∈ V (G) \ D, u is adjacent to somevertex v ∈ D. The domination number, γ(G) ofG, is the minimum cardinality of a dominating set of G. A setD ⊆ V (G) is a total dominating set if for eachu ∈ V (G), u is adjacent to some vertex v ∈ D. Thetotal domination number, γt (G) of G, is theminimum cardinality of a total dominating set o...

Let $G$ be a simple graph with vertex set $V$. A double Roman dominating function (DRDF) on $G$ is a function $f:Vrightarrow{0,1,2,3}$ satisfying that if $f(v)=0$, then the vertex $v$ must be adjacent to at least two vertices assigned $2$ or one vertex assigned $3$ under $f$, whereas if $f(v)=1$, then the vertex $v$ must be adjacent to at least one vertex assigned $2$ or $3$. The weight of a DR...

let $g=(v,e)$ be a‎ ‎simple graph of order $n$ and size $m$‎. ‎an $r$-matching of $g$ is ‎a set of $r$ edges of $g$ which no two of them have common vertex‎. ‎the hosoya index $z(g)$ of a graph $g$ is defined as the total‎ ‎number of its matchings‎. ‎an independent set of $g$ is a set of‎ ‎vertices where no two vertices are adjacent‎. ‎the ‎merrifield-simmons index of $g$ is defined as the tota...

the total irregularity of a graph g is defined as 〖irr〗_t (g)=1/2 ∑_(u,v∈v(g))▒〖|d_u-d_v |〗, where d_u denotes the degree of a vertex u∈v(g). in this paper by using the gini index, we obtain the ordering of the total irregularity index for some classes of connected graphs, with the same number of vertices.

A total Roman dominating function on a graph $G$ is a function $f: V(G) rightarrow {0,1,2}$ such that for every vertex $vin V(G)$ with $f(v)=0$ there exists a vertex $uin V(G)$ adjacent to $v$ with $f(u)=2$, and the subgraph induced by the set ${xin V(G): f(x)geq 1}$ has no isolated vertices. The total Roman domination number of $G$, denoted $gamma_{tR}(G)$, is the minimum weight $omega(f)=sum_...