<abstract><p>A neighbor sum distinguishing (NSD) total coloring $ \phi $ of $ G $ is a proper total coloring such that $ \sum_{z\in E_{G}(u)\cup\{u\}}\phi(z)\neq\sum_{z\in E_{G}(v)\cup\{v\}}\phi(z) $ for each edge $ uv\in E(G) $. Pilśniak and Woźniak asserted that each graph with a maximum degree $ \Delta $ admits an NSD total $ (\Delta+3) $-coloring in 2015. In this paper, we prove that the list version of this conjecture holds for any IC-planar graph with $ \Delta\geq10 $ but without five cycles by applying the discharging method, which improves the result of Zhang (NSD list total coloring of IC-planar graphs without five cycles).</p></abstract>
A proper total k-coloring ϕ of G with ∑z∈EG(u)∪{u}ϕ(z)≠∑z∈EG(v)∪{v}ϕ(z) for each uv∈E(G) is called a total neighbor sum distinguishing k-coloring, where EG(u)={uv|uv∈E(G)}. Pilśniak and Woźniak conjectured that every graph with maximum degree Δ exists a total neighbor sum distinguishing (Δ+3)-coloring. In this paper, we proved that any IC-planar graph with Δ≥12 satisfies this conjecture, which improves the result of Song and Xu [J. Comb. Optim., 2020, 39, 293–303].
In this paper we investigate the structures of short cycles in a weighted graph. By Thomassen's 3-path-condition theory (Thomassen in J Comb Theory Ser 48:155-177, 1990), it is easy to find a shortest cycle in a collection of cycles beyond a subspace of the cycle space of a graph. What is more difficult for one to do is to find a shortest cycle within a subspace of the cycle space in polynomial time. By using the Dijkstra's algorithm (Dijkstra in Numer Math 1:55-61, 1959) we find a collection C of cycles containing many types of short cycles within a given subspace of the cycle space of a graph and this implies a polynomial time algorithm (called extended fundamental cycle algorithm) to locate all the possible shortest cycles in a weighted graph. In the case of unweighted graphs, the algorithm may also find every shortest even cycle in a graph, this greatly improved a result of Grötschel and Pulleyblank (Oper Res Lett 1:23-27, 1981/82), Monien (Computing 31:355-369, 1983), Yuster and Zwick (SIAM J Discrete Math 10:209-222, 1997). In fact, our algorithm shows that there are at most O(n 4 ) many such short cycles in an unweighted graph of order n. Further more, our fundamental cycle method may find a minimum cycle base (or simply MCB as some scholars named) in the cycle space of a graph. Since the structure of MCB's is unique (Ren and Deng in Discrete Math 307:2654-2660, 2007, this shows that, in the sense, The research is supported by NSFC 11171114.B Fugang Chao 123 66 Graphs and Combinatorics (2016) 32:65-77cycles in a MCB are nearly-fundamental (i.e., each element in a MCB is a sum of at most two fundamental cycles). This provides a new way to study MCB.
In this paper we investigate the relation between odd components of co-trees and graph embeddings. We show that any graph G must share one of the following two conditions: (a) for each integer h such that G may be embedded on S h , the sphere with h handles, there is a spanning tree T in G such that h = 1 2 (β(G)−ω(T )), where β(G) and ω(T ) are, respectively, the Betti number of G and the number of components of G − E(T ) having odd number of edges; (b) for every spanning tree T of G, there is an orientable embedding of G with exact ω(T ) + 1 faces. This extends Xuong and Liu's theorem [5,6] to some other ( possible ) genera. Infinitely many examples show that there are graphs which satisfy (a) but (b). Those make a correction of a rseult of D.Archdeacon[2, theorem 1].
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.