Asymptotically Effective Method to Explore Euler Path in a Graph

Euler path is one of the most interesting and widely discussed topics in graph theory. An Euler path (or Euler trail) is a path that visits every edge of a graph exactly once. Similarly, an Euler circuit (or Euler cycle) is an Euler trail that starts and ends on the same node of a graph. A graph having Euler path is called Euler graph. While tracing Euler graph, one may halt at arbitrary nodes while some of its edges left unvisited. In this paper, we have proposed some precautionary steps that should be considered in exploring a deadlock-free Euler path, i.e., without being halted at any node. Simulation results show that our proposed approach improves the process of exploring the Euler path in an undirected connected graph without interruption. Furthermore, our proposed algorithm is complete for all types of undirected Eulerian graphs. The paper concludes with the proofs of the correctness of proposed algorithm and its computation complexity.

Circuit Covers of Signed Eulerian Graphs

A signed circuit cover of a signed graph is a natural analog of a circuit cover of a graph, and is equivalent to a covering of its corresponding signed-graphic matroid with circuits. It was conjectured that a signed graph whose signed-graphic matroid has no coloops has a 6-cover. In this paper, we prove that the conjecture holds for signed Eulerian graphs.

FIBONACCI AND SUPER FIBONACCI GRACEFUL LABELLINGS OF SOME TYPES OF GRAPHS

We consider the basic theoretical information regarding the Fibonacci graceful graphs. An injective function is said a Fibonacci graceful labelling of a graph of a size , if it induces a bijective function on the set of edges , where by the rule , for any adjacent vertices A graph that allows such labelling is called Fibonacci graceful. In this paper, we introduce the concept of super Fibonacci graceful labelling, narrowing the set of vertex labels, i.e. Four types of problems to be studied are selected. In the problem of the first type, the following question is raised: is there a graph that allows a certain kind of labelling, and under what conditions does this take place? The problem of the second type is the problem of construction: it is necessary, for a given system of requirements for the graph, to construct (at least one) its labelling that would satisfy this system. The following two types of problems relate to enumeration problems: for a given graph, determine the number of different Fibonacci and / or super Fibonacci graceful labellings; build all the different labellings of a given kind. As a result of solving these problems, functions were found that generate Fibonacci and super Fibonacci graceful labellings for graphs of cyclic structure; necessary and sufficient conditions for the existence of Fibonacci graceful labelling for disjunctive union of cycles, super Fibonacci graceful labelling for cycles, Eulerian graphs are obtained; the number of non-equivalent labellings of the cycle is determined; conditions for the existence of a super Fibonacci graceful labelling of a one-point connection of arbitrary connected super Fibonacci graceful graphs … …, are presented

Covers, Orientations and Factors

Given a graph $G$ with only even degrees, let $\varepsilon(G)$ denote the number of Eulerian orientations, and let $h(G)$ denote the number of half graphs, that is, subgraphs $F$ such that $d_F(v)=d_G(v)/2$ for each vertex $v$. Recently, Borbényi and Csikvári proved that $\varepsilon(G)\geq h(G)$ holds true for all Eulerian graphs, with equality if and only if $G$ is bipartite. In this paper we give a simple new proof of this fact, and we give identities and inequalities for the number of Eulerian orientations and half graphs of a $2$-cover of a graph $G$.

Fuzzy Eulerian and Fuzzy Hamiltonian Graphs with Their Applications

In this article we discussed prominence of Fuzzy Eulerian and Fuzzy Hamiltonian graphs. Fuzzy logic is introduced to study the uncertainty of the event. In Fuzzy set theory we assign a membership value to each element of the set which ranges from 0 to 1. The earnest efforts of the researchers are perceivable in the relevant establishment of the subject integrating coherent practicality and reality. Fuzzy graphs found an escalating number of applications in day to day life system where the information intrinsic in the system varies with different levels of accuracy. In this article we initiated the model of fuzzy Euler graphs (FEG) and also fuzzy Hamiltonian graphs (FHG). We explored about fuzzy walk, fuzzy path, fuzzy bridge, fuzzy cut node, fuzzy tree, fuzzy blocks, fuzzy Eulerian circuit and fuzzy Hamiltonian cycle. Here we studied some applications of Fuzzy Eulerian graphs and fuzzy Hamiltonian graphs in real life.

On the Generalized Spectral Characterizations of Eulerian Graphs

A graph $G$ is said to be determined by its generalized spectra (DGS for short) if, for any graph $H$, graphs $H$ and $G$ are cospectral with cospectral complements imply that $H$ is isomorphic to $G$. In Wang (J. Combin. Theory, Ser. B, 122 (2017) 438-451), the author gave a simple method for a graph to be DGS. However, the method does not apply to Eulerian graphs. In this paper, we gave a simple method for a large family of Eulerian graphs to be DGS. Numerical experiments are also presented to illustrate the effectiveness of the proposed method.