scholarly journals On incidence coloring of complete multipartite and semicubic bipartite graphs

2018 ◽  
Vol 38 (1) ◽  
pp. 107 ◽  
Author(s):  
Robert Janczewski ◽  
Anna Małafiejska ◽  
Michał Małafiejski
Keyword(s):  
Bipartite Graphs ◽  
Complete Multipartite ◽  
Incidence Coloring

scholarly journals Biconed Graphs, Weighted Forests, and $h$-Vectors of Matroid Complexes

10.37236/9849 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Preston Cranford ◽  
Anton Dochtermann ◽  
Evan Haithcock ◽  
Joshua Marsh ◽  
Suho Oh ◽  
...  
Keyword(s):  
Bipartite Graphs ◽  
Nonzero Entry ◽  
Combinatorial Interpretation ◽  
Underlying Graph ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Complete Bipartite Graphs ◽  
Ferrers Graphs ◽  
Independence Complex ◽  
Complete Multipartite

A well-known conjecture of Richard Stanley posits that the $h$-vector of the independence complex of a matroid is a pure ${\mathcal O}$-sequence. The conjecture has been established for various classes but is open for graphic matroids. A biconed graph is a graph with two specified 'coning vertices', such that every vertex of the graph is connected to at least one coning vertex. The class of biconed graphs includes coned graphs, Ferrers graphs, and complete multipartite graphs.  We study the $h$-vectors of graphic matroids arising from biconed graphs, providing a combinatorial interpretation of their entries in terms of '$2$-weighted forests' of the underlying graph. This generalizes constructions of Kook and Lee who studied the Möbius coinvariant (the last nonzero entry of the $h$-vector) of graphic matroids of complete bipartite graphs. We show that allowing for partially $2$-weighted forests gives rise to a pure multicomplex whose face count recovers the $h$-vector, establishing Stanley's conjecture for this class of matroids.  We also discuss how our constructions relate to a combinatorial strengthening of Stanley's Conjecture (due to Klee and Samper) for this class of matroids.

scholarly journals Partitioning 3-Edge-Colored Complete Equi-Bipartite Graphs by Monochromatic Trees under a Color Degree Condition

10.37236/855 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Xueliang Li ◽  
Fengxia Liu
Keyword(s):  
Bipartite Graphs ◽  
Complete Multipartite Graph ◽  
Colored Graph ◽  
Degree Condition ◽  
Partition Number ◽  
Color Degree ◽  
Minimum Integer ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
Vertex Disjoint

The monochromatic tree partition number of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum integer $k$ such that whenever the edges of $G$ are colored with $r$ colors, the vertices of $G$ can be covered by at most $k$ vertex-disjoint monochromatic trees. In general, to determine this number is very difficult. For 2-edge-colored complete multipartite graph, Kaneko, Kano, and Suzuki gave the exact value of $t_2(K(n_1,n_2,\cdots,n_k))$. In this paper, we prove that if $n\geq 3$, and $K(n,n)$ is 3-edge-colored such that every vertex has color degree 3, then $t_3(K(n,n))=3$.

scholarly journals Logics of bipartite graphs and complete multipartite graphs

1982 ◽  
Vol 107 (4) ◽  
pp. 425-427
Author(s):  
Zdeněk Andres ◽  
Bohdan Zelinka
Keyword(s):  
Bipartite Graphs ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Complete Multipartite

Incidence Coloring Number of Some Join Graphs

2014 ◽  
Vol 602-605 ◽  
pp. 3185-3188
Author(s):  
Gui Xiang Dong ◽  
Xiu Fang Liu
Keyword(s):  
Bipartite Graphs ◽  
Complete Graphs ◽  
Coloring Number ◽  
Complete Bipartite Graphs ◽  
Incidence Coloring Number ◽  
Incidence Coloring

The incidence coloring of a graph is a mapping from its incidence set to color set in which neighborly incidences are assigned different colors. In this paper, we determined the incidence coloring numbers of some join graphs with paths and paths, cycles, complete graphs, complete bipartite graphs, respectively, and the incidence coloring numbers of some join graphs with complete bipartite graphs and cycles, complete graphs, respectively.

scholarly journals Bipartite Graphs whose Squares are not Chromatic-Choosable

10.37236/4343 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Seog-Jin Kim ◽  
Boram Park
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Bipartite Graphs ◽  
Total Coloring ◽  
Natural Question ◽  
List Total Coloring ◽  
Multipartite Graphs ◽  
Complete Multipartite ◽  
List Chromatic Number ◽  
Total Coloring Conjecture

The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. Let $\chi(H)$ and $\chi_{\ell}(H)$ be the chromatic number and the list chromatic number of $H$, respectively. A graph $H$ is called chromatic-choosable if $\chi_{\ell} (H) = \chi(H)$. It is an interesting problem to find graphs that are chromatic-choosable.Motivated by the List Total Coloring Conjecture, Kostochka and Woodall (2001) proposed the List Square Coloring Conjecture which states that $G^2$ is chromatic-choosable for every graph $G$. Recently, Kim and Park showed that the List Square Coloring Conjecture does not hold in general by finding a family of graphs whose squares are complete multipartite graphs and are not chromatic choosable. It is a well-known fact that the List Total Coloring Conjecture is true if the List Square Coloring Conjecture holds for special class of bipartite graphs. Hence a natural question is whether $G^2$ is chromatic-choosable or not for every bipartite graph $G$.In this paper, we give a bipartite graph $G$ such that $\chi_{\ell} (G^2) \neq \chi(G^2)$. Moreover, we show that the value $\chi_{\ell}(G^2) - \chi(G^2)$ can be arbitrarily large.

scholarly journals Interval incidence coloring of bipartite graphs

2014 ◽  
Vol 166 ◽  
pp. 131-140 ◽  
Author(s):  
Robert Janczewski ◽  
Anna Małafiejska ◽  
Michał Małafiejski
Keyword(s):  
Bipartite Graphs ◽  
Incidence Coloring

Bipartite Graphs and their Applications

1998 ◽  
Author(s):  
Armen S. Asratian ◽  
Tristan M. J. Denley ◽  
Roland Häggkvist
Keyword(s):  
Bipartite Graphs

Complete Multipartite Graph Codes

2018 ◽  
Author(s):  
Yuta Kumano ◽  
Yoshiyuki Sakamaki ◽  
Hironori Uchikawa
Keyword(s):  
Complete Multipartite Graph ◽  
Graph Codes ◽  
Multipartite Graph ◽  
Complete Multipartite
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