1-planar graphs are odd 13-colorable 2023, Discrete Mathematics Citation Excerpt : We improved this result to 13. (Claim 2 in [13]) Every vertex of odd degree in G has degree at least 7. Our key contribution is the following lemma, which is a strengthening of Lemma 2.1 in [10]. Sho...
A graph is 1-planar if it can be drawn in the Euclidean plane so that each edge is crossed by at most one other edge. A 1-planar graph on n vertices is optimal if it has 4n 8 edges. We prove that 1-planar graphs with girth at least 6 are (1,1,1,1)-colorable (in the ...
Since planar graphs are 4-colorable, see [[1], [2]], we have L1,0(G)≤3 for every planar graph G. L(0,1)-labelings of cycles and trees were studied by Bertossi and Bonuccelli [4]. They proved that L0,1(Cn)=2 if n≡0(mod4); otherwise L0,1(Cn)=3 for any n-vertex ...
Let be the family of planar graphs without 3-cycles adjacent to cycles of length 3 or 5. This paper proves that everyone in is (3,1)-colorable. This is the best possible in the sense that there are members in which are not (3,0)-colorable.Zhengke,Miao...
We prove in this paper that such graphs are (1,1,0)(1,1,0)-colorable.Runrun LiuXiangwen LiGexin YuDiscrete MathematicsRunrun LIU, Xiangwen LI, Gexin YU. Planar graphs without 5-cycles and intersecting triangles are (1, 1, 0)- colorable. Eprint Arxiv, 2014....
For a number ≥2 $\\ell \\ge 2$, let G ${{\\mathscr{G}}}_{\\ell }$ denote the family of graphs which have girth 2 1 $2\\ell 1$ and have no odd hole with length greater than 2 1 $2\\ell 1$. Wu et al. conjectured that every graph in ≥2G ${\\bi...
Hill O, Smith D, Wang Y, Xu L, Yu G (2013) Planar graphs without 4-cycles or 5-cycles are (3, 0, 0)- colorable. Discret Math 313:2312-2317.O. Hill, D. Smith, Y. Wang, L. Xu, G. Yu, Planar graphs without 4-cycles or 5-cycles are ( 3 , 0 , 0 ) -colorable, ...
Borodin, Montassier and Raspaud asked: Is every planar graph without adjacent cycles of length at most five 3-colorable, i.e., (0, 0, 0)-colorable? This problem has now been answered negatively by Cohen-Addad et al. who successfully constructed a non-3-colorable planar graph with ...
PlanargraphsImpropercoloringDischargingmethodA graph G is \\((d_1, d_2)\\)-colorable if its vertices can be partitioned into subsets \\(V_1\\) and \\(V_2\\) such that in \\(G[V_1]\\) every vertex has degree at most \\(d_1\\) and in \\(G[V_2]\\) every vertex ...
Graphs that embedded in any fixed surfaces with sufficiently large maximum degree Δ is total-(Δ 1)-colorabledoi:10.3934/math.2024075Qiming FangLi ZhangAIMS Mathematics