A closed walk in a connected graph G that contains every edge of G exactly once is an Eulerian circuit. A graph is Eulerian if it contains an Eulerian circuit. It is well known that a connected graph G is Eulerian if and only if every vertex of G is even. An Eulerian walk in a connected graph G is a closed walk that contains every edge of G at least once, while an irregular Eulerian walk in G is an Eulerian walk that encounters no two edges of G the same number of times. The minimum length of an irregular Eulerian walk in G is called the Eulerian irregularity of G and is denoted by EI(G). It is known that if G is a nontrivial connected graph of size m, then [...] . A necessary and sufficient condition has been established for all pairs k,m of positive integers for which there is a nontrivial connected graph G of size m with EI(G) = k. A subgraph F in a graph G is an even subgraph of G if every vertex of F is even. We present a formula for the Eulerian irregularity of a graph in terms of the size of certain even subgraph of the graph. Furthermore, Eulerian irregularities are determined for all graphs of cycle rank 2 and all complete bipartite graphs
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In a red-blue coloring of a nonempty graph, every edge is colored red or blue. If the resulting edge-colored graph contains a nonempty subgraph G without isolated vertices every edge of which is colored the same, then G is said to be monochromatic. For two nonempty graphs G and H without isolated vertices, the mono- chromatic Ramsey number mr(G,H) of G and H is the minimum integer n such that every red-blue coloring of Kn results in a monochromatic G or a monochromatic H. Thus, the standard Ramsey number of G and H is bounded below by mr(G,H). The monochromatic Ramsey numbers of graphs belonging to some common classes of graphs are studied. We also investigate another concept closely related to the standard Ram- sey numbers and monochromatic Ramsey numbers of graphs. For a fixed integer n ≥ 3, consider a nonempty subgraph G of order at most n con- taining no isolated vertices. Then G is a common monochromatic subgraph of Kn if every red-blue coloring of Kn results in a monochromatic copy of G. Furthermore, G is a maximal common monochromatic subgraph of Kn if G is a common monochromatic subgraph of Kn that is not a proper sub- graph of any common monochromatic subgraph of Kn. Let S(n) and S*(n) be the sets of common monochromatic subgraphs and maximal common monochromatic subgraphs of Kn, respectively. Thus, G ∈ S(n) if and only if R(G,G) = mr(G,G) ≤ n. We determine the sets S(n) and S*(n) for 3 ≤ n ≤ 8.
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A twin edge k-coloring of a graph G is a proper edge coloring of G with the elements of Zk so that the induced vertex coloring in which the color of a vertex v in G is the sum (in Zk) of the colors of the edges incident with v is a proper vertex coloring. The minimum k for which G has a twin edge k-coloring is called the twin chromatic index of G. Among the results presented are formulas for the twin chromatic index of each complete graph and each complete bipartite graph
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