The main result of this paper completely settles Bermond's conjecture for bipartite graphs of odd degree by proving that if G is a bipartite (2k+1)-regular graph that is Hamilton decomposable, then the line graph, L(G), of G is also Hamilton decomposable. A similar result is obtained for 5-regular graphs, thus providing further evidence to support Bermond's conjecture.

Extending the work of K. L. Collins and A.N. Trenk, we characterize connected bipartite graphs with large distinguishing chromatic number. In particular, if G is a connected bipartite graph with maximum degree ∆ ≥ 3, then χD(G) ≤ 2∆ − 2 whenever G 6∼= K∆−1,∆, K∆,∆.

Some bipartite Hamilton decomposable graphs that are regular of degree δ ≡ 2 (mod 4) are shown to have Hamilton decomposable line graphs. One consequence is that every bipartite Hamilton decomposable graph G with connectivity κ(G) = 2 has a Hamilton decomposable line graph L(G).

Given a graph L, in this article we investigate the anti-Ramsey number χS (n,e,L), defined to be the minimum number of colors needed to edge-color some graph G(n,e) with n vertices and e edges so that in every copy of L inG all edges have different colors. We call such a copy of L totally multicolored (TMC). In [7] among many other interesting results and problems, Burr, Erdős, Graham, and T. S...

A bipartite graph G = (L,R;E) with at least one edge is said to be identifiable if for every vertex v ∈ L, the subgraph induced by its non-neighbors has a matching of cardinality |L| − 1. An `-subgraph of G is an induced subgraph of G obtained by deleting from it some vertices in L together with all their neighbors. The Identifiable Subgraph problem is the problem of determining whether a given...

Bruce Reed asks the following question: Can we determine whether a bipartite graph contains a chordless cycle whose length is a multiple of 4? We show that the two following more general questions are equivalent and we provide an answer. Given a bipartite graph G where each edge is assigned a weight + 1 or I, l determine whether G contains a cycle whose weight is a multiple of 4, l determine wh...

We first show that a graph is not bipartite if it contains any cycles of odd length. Starting at an arbitrary vertex u in this cycle, we note that there is a path of length 2k + 1 from u to v. We also note that to be bipartite, we must alternate placing vertices on this path into the partitions L and R. However, if we place u in L, then, we see that after following the path along 2k + 1 edges, ...

Given a graph G = (V, E), we define the transpose degree sequence d T j to be equal to the number of vertices of degree at least j. We define L G , the graph Laplacian, to be the matrix, whose rows and columns are indexed by the vertex set V , whose diagonal entry at v is the degree of v and whose value at a pair (v, w) is −1 if (v, w) ∈ E and 0 otherwise. Grone and Merris conjectured [GM] Conj...

This paper shows that with B = {1, 2, . . . , n}, the smallest k such that (B ×B)− {(j, j) | j ∈ B} = k ⋃ i=1 (Ci ×Di) is s(n), where s(n) is the smallest integer k such that n 6 ( k b k 2 c ) . This provides a simple set-based formulation and a new proof of a result for boolean ranks [2] and biclique covering of bipartite graphs [1, 5], making these intricate results more accessible.