In this paper we introduce remainder cordial labeling of graphs. Let $G$ be a $(p,q)$ graph. Let $f:V(G)rightarrow {1,2,...,p}$ be a $1-1$ map. For each edge $uv$ assign the label $r$ where $r$ is the remainder when $f(u)$ is divided by $f(v)$ or $f(v)$ is divided by $f(u)$ according as $f(u)geq f(v)$ or $f(v)geq f(u)$. The function$f$ is called a remainder cordial labeling of $G$ if $left| e_{...

The vertices of the odd-distance graph are the points of the plane R. Two points are connected by an edge if their Euclidean distance is an odd integer. We prove that the chromatic number of this graph is at least five. We also prove that the odd-distance graph in R is countably choosable, while such a graph in R is not.

We conjecture that any 2-regular simple graph has an SSA labeling. We provide several special cases to support our conjecture. Most of our constructions are based on Skolem sequences and on an extension of it. We establish a connection between simply sequentially additive labelings of 2-regular graphs and ordered graceful labelings of spiders.

‎in this paper we introduce the concept of order difference interval graph $gamma_{odi}(g)$ of a group $g$‎. ‎it is a graph $gamma_{odi}(g)$ with $v(gamma_{odi}(g)) = g$ and two vertices $a$ and $b$ are adjacent in $gamma_{odi}(g)$ if and only if $o(b)-o(a) in [o(a)‎, ‎o(b)]$‎. ‎without loss of generality‎, ‎we assume that $o(a) leq o(b)$‎. ‎in this paper we obtain several properties of $gamma_...

A graph G is said to have a totally magic cordial labeling with constant C if there exists a mapping f : V (G) ∪ E(G) → {0, 1} such that f(a) + f(b) + f(ab) ≡ C (mod 2) for all ab ∈ E(G) and |nf (0) − nf (1)| ≤ 1, where nf (i) (i = 0, 1) is the sum of the number of vertices and edges with label i. In this paper, we give a necessary condition for an odd graph to be not totally magic cordial and ...

A graph G(p, q) is said to be odd harmonious if there exists an injection f : V (G)→ {0, 1, 2, · · · , 2q − 1} such that the induced function f∗ : E(G) → {1, 3, · · · , 2q − 1} defined by f∗(uv) = f(u) + f(v) is a bijection. A graph that admits odd harmonious labeling is called odd harmonious graph. In this paper we prove that any two even cycles sharing a common vertex and a common edge are od...

A graph G(p, q) is said to be odd harmonious if there exists an injection f : V (G) → {0, 1, 2, · · · , 2q − 1} such that the induced function f∗ : E(G) → {1, 3, · · · , 2q − 1} defined by f∗(uv) = f(u) + f(v) is a bijection. A graph that admits odd harmonious labeling is called odd harmonious graph. In this paper, we prove that shadow and splitting of graph K2,n, Cn for n ≡ 0 (mod 4), the grap...

A totally odd H-subdivision means a subdivision of a graph H in which each edge of H corresponds to a path of odd length. Thus this concept is a generalization of a subdivision of H. In this paper, we give a structure theorem for graphs without a fixed graph H as a totally odd subdivision. Namely, every graph with no totally odd H-subdivision has a tree-decomposition such that each piece is eit...