Asian Journal of Pure and Applied Mathematics

Let η≥3 be an odd integer. For a simple connected graph G of order n, a bijective function f:V(G)→{1,2,…,n} is called Euler cordial labeling modulo η if the induced function f_η^*:E(G)→{0,1}, defined by f_η^* (uv)=0 whenever \(\frac{[f(a)+f(b)]^ϕ(η) -1}{n}\) is even or f(a)+f(b) and η are not relatively prime, and f_η^* (uv)=1 whenever \(\frac{[f(a)+f(b)]^ϕ(η) -1}{n}\) is odd, satisfies the condition |e_fη^* ) (0)-e_fη^* ) (1)|≤1 where e_fη^* ) (i) is the number of edges with label i (i=0,1). This study explores the Euler cordial labeling of path graphs, star graphs, complete bipartite graphs, bistar graphs, alternate cycle snake graphs, m-tuple star graphs, and book graphs. The path graphs P_η and P_2η, the alternate cycle snake graph A_m (C_η ) (if gcd⁡((η+3)/2,η)= 1), and the book graph B_η,2q (if gcd⁡((η+5)/2,η)=1) were determined to admit Euler cordial labeling modulo η, where η is an odd integer with η≥3. Moreover, the star graphs S_p and S_2p-1, the complete bipartite graph K_q(p-1),_q, the bistar graph B_p,p, and the m-tuple star graph S_m,p admit Euler cordial labeling modulo p, where p is an odd prime.