R. Sampathkumar

Extremal graphs in some coloring problems

Abstract For a simple graph G with chromatic number χ ( G ), the Nordhaus-Gaddum inequalities give upper and lower bounds for χ ( G ) χ ( G c ) and χ ( G ) + χ ( G c ). Based on a characterization by Fink of the extremal graphs G attaining the lower bounds for the product and sum, we characterize the extremal graphs G for which A ( G ) B ( G c ) is minimum, where A and B are each of chromatic number, achromatic number and pseudoachromatic number. Characterizations are also provided for several cases in which A ( G ) + B ( G c ) is minimum.

Cyclic orthogonal double covers of 4-regular circulant graphs

An orthogonal double cover (ODC) of a graph H is a collection G = { G v : v � V ( H ) } of | V ( H ) | subgraphs of H such that every edge of H is contained in exactly two members of G and for any two members G u and G v in G , | E ( G u ) � E ( G v ) | is 1 if u and v are adjacent in H and it is 0 if u and v are nonadjacent in H . An ODC G of H is cyclic (CODC) if the cyclic group of order | V ( H ) | is a subgroup of the automorphism group of G . In this paper, we are concerned with CODCs of 4-regular circulant graphs. Highlights� Let G be a simple graph with four edges and let H be a 4-regular circulant graph.�Problem: Does there exists a cyclic orthogonal double cover of H by G ? � In this study, we have completely settled this problem. � We use a special kind of labelling for its proof.

On the existence of graphs with prescribed coloring parameters

Abstract The pseudoachromatic number ψ(G) of a graph G is the maximum size of a vertex partition of G (where the sets of the partition may or may not be independent) such that between any two distinct parts, there is at least one edge of G . Here, we prove that if 2⩽a⩽b⩽c , then there exists a graph G with chromatic number a , achromatic number b and pseudoachromatic number c .

Decompositions of regular graphs into K c n ∨2K 2

Abstract The join Knc ∨ 2K2 is the graph obtained by taking a copy of Knc and two disjoint copies of K2, disjoint from Knc, and joining every vertex of Knc to every vertex of 2K2. In this paper we show that for each positive integer n, the graph Knc ∨ 2K2 admits a p-valuation and has gracefulness 4n + 3. Further, for any finite set S0 of positive integers and n ⩾ 2Max {x ∈ ϵ S0} + 1, a new graph valuation, ρ(n; S0), is introduced. Finally, some open problems are proposed.

Modular colorings of join of two special graphs

For k≥2, a modular k-coloring of a graph G without isolated vertices is a coloring of the vertices of G with the elements in Zk having the property that for every two adjacent vertices of G, the sums of the colors of their neighbors are different in Zk. The minimum k for which G has a modular k-coloring is the modular chromatic number of G. In this paper, we determine the modular chromatic number of join of two special graphs.

Twin edge colorings of certain square graphs and product graphs

A twin edge

Orthogonal double covers of complete bipartite graphs.

k\!

Graphs with vertex-coloring and detectable 2-edge-weighting

-coloring of a graph

Mutually orthogonal graph squares

G

Eulerian Cycle Decomposition Conjecture for the line graph of complete graphs

is a proper edge